Thank you all for coming down this morning, I think we've got a busy agenda, so we better get at it. First I'd like to ask for an update on the tax reform conference report, in order that I might have some idea of what to expect of the bill that would get to my desk. I hope that you will act expeditiously in completing action on that bill. I want you to know that the bill that has emerged from conference meets the objectives I outlined in my letters to Bob Marple and Jack Kent last December. In that correspondence, I said that I'd only sign a bill if it allows for a full 2,000-dollar personal exemption for both itemizers and non-itemizers, basic tax incentives for American industries. I think it dates so as to the race act of action in the minds of those in this plan for future annual investment, and a minimum tax which allows no individual enterprise to escape paying a fair share of the overall tax burden, a rate structure with a maximum rate for having the proposal of the tax practice that reduced taxes for middle-income working Americans. And I asked those of you in the House last December to vote for that House bill as an expression of your faith that we could then get a good bill in the Senate through conference. I think that was something of a record. I don't know that anyone's ever said to you, I want you to sign a bill with our veto if it gets to my desk the way it is, but that was what we did. But with your leadership and hard work that's been done, and I'm satisfied that we now have a bill that meets our objectives and should become law, and I want to ask Jim Baker if there's anything he wants to add to this. Well, nothing, Mr. President, other than that we're... My understanding of this was much more limited, but part of becoming a California part of the process is developing a love and appreciation for that part of America that is uniquely Hispanic. In California that includes our many Americans of Hispanic descent, with courage above and beyond the call of duty, distinguished themselves in battle, defending our liberty. He penned those words when they were considered a discrimination by any trace. It resolved to maintain dignity even in times of great adversity. Never was that brought home to me more. There, close to the highway, and one elderly gentleman invited me in to his house, this house. The furniture there it was, standing, was sitting in all of that mud. And as we stood there in the living room, the administration has been to expand opportunity to all Americans. And if I read the Hispanic community right, that's all that's expected. I've always found it strange that the idea that people should work hard to support their families and improve their well-being is labeled the Protestant work ethic. A man who early in his life picked potatoes for a living, with freedom and opportunity available to him, with hard work and a commitment to actions. He is today a successful businessman and resident of the United States. After I understand that the Hispanic Chambers Convention will open tomorrow, I had no previous experience in that job.  It's wonderful sometimes to not know the things you can't do 8%. And I should say there will be three brackets. Because many lower income people, 6 million we estimate, will be taken off the roads all together. Their bracket will be zero. So that's a third bracket, zero. This bill, which we fought so hard to get... What just happened, Donald's corporation along with the Hispanic caucus launched a dark contest. The theme was what my spending heritage meant to know. And Monica just had it with him. Until I continue in the right direction to shut the government down, you're going to get one you like. I don't get one that I think is good for the people, so I'm going to help with what we're trying to do to get out of here. Well, you said you'd drawn a line in the sand on this business with the house. I was compromised on South African sanctions with the president. What about South African? I have sent some information on that after the deal with the man. My only query is, I did not really unload here. This is a full report. Well, I thank you very much. I will wait just for a second. I have something I have to... I'll put the bears on the high school essay contest. Yes, listen, I think very highly of that. I think it's a great idea. I like your idea about the celebration being kind of a silly question. I think our people need separation. Yes. Including all present company, Mr. President. I agree. Happy birthday. Happy birthday. Happy anniversary, Mr. Thirty-Night. That's a good one. Wait a minute. Don't go too far. I'm up to 36 already. Let's stop the talk. All right. Alan, I'm delighted to hear your reaction about that high school essay contest. Even if we write off the people on the edge of you and Donald, and all the rest of the season, these folks... And in recent years, their school has given them much in the America. Come on, make a wish and blow those out. Oh, thank you very much. No role in the investigation. Have you run a room or a staff room? I need to see the other person in the room. I need to see if he works here in the building. Oh, you work for Hailey Barber, yeah. I do something every day down there. I'm just sitting in this room. I'm just going to get these shutters. Peter? There's only one day it just seemed like that. For an athlete, I was an analyst for, we had many of the commissioners. And celebrated taking note of the original Anapolis Convention with Governor Hughes and Governor Calvary which was the first state that ratified the Constitution Then I got that. Was that the convention that Madison had really worked over? They had really, right? There's a quality connection to commercial matters, a meeting on commercial matters. But they knew what they were shooting for. They were shooting for a constitutional convention. They got it. It was very interesting. Our commission meetings took up long hours so we'd slip out and take a part in some event that they had tied in. Firstly, this is the first 13 states making a great deal out of it. If you get in the villain don't it? The heart service. Got a mom living with me today, Mr. President? Well, that would probably be mom. Well, that's a great one. My daughter, Kirsten. That's my mother, Esther Swann. My wife, Jill. Yeah, there. Nice to see you again. Good to see you. I think you've got a rather nice over here. Well, we'll come here and have a family group. That's great. Right? Well, the way you do it is you put half of them. You put one side of the president, half on the other side. Okay, now one more person over on this side. Okay. Well, that one's gone. Okay, thanks. It's off. Now we can say it's all Don's idea. Yeah. I guess we're waiting for someone there to go. Yes, sir. We certainly do. Just a few souvenirs. Thank you. A couple things for you. A couple things. Thank you very much. And these were the silver and a cube. Thank you. But she's not old enough to drive yet. So fix them up. We'll just have a cube. We'll only get it in the house. All right. Thank you. This is a real pleasure. I appreciate it, Mr. Brick. You guys better go on. You guys better go on. You guys better go on. What's everybody doing? What's everybody doing? Well, I think we know why we're all here, and I think we all really should be smiling through our tears. But after 18 years, I can't deny it. All the way back in Sacramento, where he was a very big part of some things that we were very proud of there, like our first comprehensive reform of welfare in the United States. And then on here in 1981, the jobs that he's had here, and I know we're going to all miss him, but I also know that we all wish him the very best and wish him well. And to that guy, there are no words to express my gratitude for all that you've meant to this administration. And many times, when I was kind of fumbling around to get an attitude on something or other and you would volunteer something, and I didn't go any further. I knew I'd have some horse's mouth. There's a pony in there somewhere. It was a thoroughbred horse that I rode. I quit. I would like to say just a few things, Mr. President. This is one of the most difficult moments in my life. But the times come, and you have been most gracious about it. I appreciate that. A lot of people are asked when they leave the White House, well, what do you miss the most? And I will miss the man. I will miss you, Ronald Reagan, President of the United States. I will miss your friendship. I'll miss your jokes. I'll miss your leadership. And I'll miss the personal example that you have set for me throughout most of my adult life. And I thank you very much for the honor and the opportunity. Thank you. Let's eat, drink, and be married. Well, thank you. Thank you. Well, let me see you. So come back and visit everyone. Oh, that's a nice thing. What a nice thing. Mr. President, and my father, and whoever. Mr. President, I see you. Well, that's good news. Wherever you are. Hi. Oh, there are a lot of them that go back to the early days. All right. Well, thank you. All right. Thank you. Bye-bye. That's very emotional. Thank you.


In this presentation, we will take a look at multiple choice questions related to corporations. First question. Number of shares a corporation can sell, a issued stock, b outstanding stock, c common stock, d preferred stock, and e authorized stock. So we'll go through this again and see if we can eliminate some options with the process of elimination question. Number of shares a corporation can sell, a issued stock. So the stock that is issued, and again, we're going to try to go through these and just see if we can eliminate some that would be more obviously not part of the problem in order to go through the process of elimination. Now the issued stock may not be a term that we fully, you know, understand. So we could keep that for now and say issued stock outstanding stock. The number of shares a corporation can sell. So if they're outstanding, that, you know, that might tell us that they haven't been sold. They're outstanding. They're out in the market. So that really doesn't tell us just by the terminology that what could be sold and basically by the terminology looks like what has been sold. So I don't think that outstanding stock would be it common stock. Now the common stock is basically just a kind of stock. It doesn't really tell us the number of the common stock, which would typically be what we're talking about that could be distributed. So common stock doesn't really tell us how many shares could be distributed. D says preferred stock. And again, that's another kind of stock. Typically we're talking about common stock when talking about numbers of shares distributed. But preferred stock doesn't tell us number of shares by that name. You would think that's just telling us what type of stock we're talking about. So I'd say we can eliminate D and E says authorized stock. And that sounds like some kind of legal term that could give us some authorization for the issuance of stock. So let's with A and E, let's go through this again and we'll pick the final answer. Question number of shares a corporation can sell is either A or E either issued stock or authorized stock. So between those two, of course, the issued stock would mean that we already sold them. They've been or they're issued. So the authorized stock, on the other hand, is what we could issue, what we could sell. So this is the authorized stock, in other words, is not what's outstanding there. That's what could be outstanding. But we have the legal right to be outstanding at this point. Outstanding is what's actually in the market at this time. So the correct answer is E. So one last time, number of shares a corporation can sell, E authorized stock. Next question. Corporate Board of Directors, A are voted on by management, B run the normal corporate activities, C are the stockholders, D are government agents, and E have final authority for managing corporate activities. So we will go through this again with the process of elimination. Corporate Board of Directors, A are voted on by management. So if we don't really understand the Board of Directors, we might think maybe management votes for the Board of Directors, maybe management is the one that does that. So I'll keep that for now. B says run the normal corporate activities. Now the Board of Directors doesn't normally run the corporate activities. That's usually run by management. So the day-to-day, meaning the day-to-day work is done by the employees, employees including the upper management, which would then hire employees to run the day-to-day operations. So I'd say the Board of Directors now not really running the normal day-to-day stuff. C are the stockholders. And we might think that, so I'll keep that for now. And D says our government agents. Now the Board of Directors doesn't necessarily need to be government agents. So the government may regulate with like the SEC Securities Exchange Commission or whatnot, but not really the Board of Directors. Those are not government agents, hopefully. E says have final authority for managing corporate activities. And that seems, you know, like reasonable too. But also when we look at these types of problems, if there's one answer that seems more, you know, longer and more detailed, not that there's anyone that's too long here or too detailed, I would say that you may want to give that one more consideration, because oftentimes a longer, more detailed answer is trying to hedge any type of wrong statement in it, meaning it's trying to be as specific as possible in order to be correct. So if there's a longer statement that seems more defined, then that may be one that you would consider more so if we didn't know what the answer was. So if we go through this again, Corporate Board of Directors, either AC or E, A are voted on by management, C are the stockholders, and E have final authority for managing corporate activities. So if we look at A and C are voted on by management are the stockholders. If you look at those two, you might start to think, who are the stockholders? Who are the Board of Directors who are management? And when we start to piece this together, the stockholders are the owners. The stockholders then are in ultimate control, but they are so due by voting on the Board of Directors and then Board of Directors hires management. So it's not the case that management hires the Board of Directors or votes on the Board of Directors. The Board of Directors assigns management. And the stockholders are the ones that are going to vote on the Board of Directors. So that means that the Board of Directors are not necessarily stockholders. The stockholders are going to be the ones that determine who is on the Board of Directors and therefore will act as agents for the best interest of the stockholders. And then E says have final authority for managing corporate activities. And that's ultimately the case, right? So the day-to-day activities are going to be done by management and the Board of Directors has the final authority because they're the ones that are responsible for putting management in place and making those big type of overarching decisions rather than the day-to-day decisions. So one more time, Corporate Board of Directors E have final authority for managing corporate activities.


Welcome, welcome. Here we are again. This is our fifth year first African landing day that will be taking place August 26 Put the date on your calendar. We're going to be celebrating our resilience We're going to be celebrating going through the fire This is the year that we celebrate every year coming and bringing our people together and this year We're going to have a keynote speaker Dr. Wanda heading Grant who will be enlightening not just the community, but everyone who participate and show up So please don't forget that the Vermont Racial Justice Alliance is The one who's been spearheading this Commencement that we're having and I want you to make sure that you all be there peace and blessings


Okay. So we're going to be talking about equilibrium. We're going to have a little bit more to say about mu because mu is important and then I'm going to talk about the bow and then I'm going to talk about the equilibrium constant. This is chapter 17 material. We've already tunneled through chapter 16, leaving out lots of stuff along the way. Non-ideal solutions, we don't have time to talk about them. They're very important and there's lots of other stuff there as well. We're already in chapter 17. How do individual reactants and products contribute to G? There could be a whole slew of reactants here. There could be a whole slew of products. How do we attribute the G for a reaction to individual reactants and products? As chemists this is obviously an important thing for us. We talked about this thought experiment right here. We have two isotopes of hydrogen in two different bulbs separated by a valve that's initially closed. Then we open it and when we open it we can describe the change in the free energy on the left side in terms of what's happening with component 1 and what's happening with component 2. In this case component 1 is H2 and component 2 is D2. So we can write two equations for the left-hand side, two equations for the right-hand side, two equations. Two terms. Here's the left-hand side, here's the right-hand side, here's component 1, here's component 2. Same thing over here. On the right-hand side component 1 and in component 2. And then if we consider that anything that leaves the right-hand side has to enter the left-hand side and anything that leaves the left-hand side has to enter the right-hand side we can write a second pair of equations, these two guys. All right and what this says is we're only talking about component 1 here, let's say hydrogen, the partial derivative of the free energy on the left side and the partial derivative of the free energy on the right side, those two things have to be equal to one another at equilibrium. And the same thing has to be true in terms of component 2. All right, so this could be for hydrogen, this could be for deuterium, these partial derivatives have to be satisfied, these equalities that involve these partial derivatives. All right, so these are so-called partial molar quantities. Whenever you take the partial derivative of a thermodynamic state function with respect to the number of moles of something it's called a partial molar quantity. Molar because you're taking the partial derivative with respect to the number of moles, okay, and the chemical potential is a special case of a partial molar quantity that involves the Gibbs energy, right, the Gibbs energy, the partial of the Gibbs energy with respect to component 1 or component 2 or component 3 would be the chemical potential of component 1 or component 2 or component 3. Mu describes how G is affected by changes in the amount of one component, right, it is in effect the G of a particular component. It is the Gibbs energy of a single component, all right. Why didn't they just call it the Gibbs energy of component 1? I don't know, it would have been a lot easier because that's what it is, right? It's just the Gibbs energy of component 1, hydrogen or deuterium. Now, it's not obvious to me when I look at this partial derivative it looks a lot more complicated than just the Gibbs energy of component 1. Why did they write it this way? This really is the Gibbs energy of component 1, all right. Do other people have a problem seeing that? Because I know I always have. So the way I like to think about it is shown here, all right. Here's a system that contains three different components and the total G for this system, think about it as just being a balance and there's a scale here, we call this the G meter. We know the free energy, excuse me, the Gibbs energy is an extensive quantity, don't we? All right, and so this is a beautiful analogy for that. We're weighing these components, it's like a G meter, this is an extensive, this is going to be an extensive quantity, isn't it? It's just like mass, okay, and so what we said is the partial derivative of G with respect to N and here we're going to pay attention to N blue, all right. We're going to try and think about the chemical potential of blue here, all right, at constant green, orange, P and T, all right. And the way to think about that derivative is that if you remove a little bit of blue or add to it, a unit of blue, here we've subtracted a unit of blue, this partial derivative is just this difference and what we're left with is the block that used to be here, that's the chemical potential of the blue. For me it's a lot easier to think about it this way, all right. Now it's clear, yes, this is just the chemical potential of the blue, all right. Now it's not the chemical potential of the blue always, it's the chemical potential of blue exactly when the composition of the system is as I'm showing it here. There's green here, there's orange here, and there's a bunch of other blue here, right. The chemical potential I'm talking about is specific to the rest of the stuff being there too. The chemical potential could be different if there was different amounts of stuff here, right. It would be as though this balance could be different, this arrow could point in different directions even for a single blue block if the rest of these guys, maybe if there's only three green guys or six orange guys, the chemical potential could change a little bit, but we're always talking about the chemical potential of this pure component that we're talking, whatever it is that we're talking about, in this case it's the blue blocks. Does this make any more sense? I don't know, it does for me. Okay, so what do we need to know about the chemical potential? It's extensive just like G. It's got units of energy per mole just like G. It's pressure dependence, it's the same as G, and so is the temperature dependence. Okay, so I can say the partial derivative of the chemical potential of any component with respect to its partial pressure is just equal to the molar volume of that component, and if I integrate that I'm going to get an equation that looks just like the equation that we write for G, except now it pertains to a single component. And the same thing with the entropy. But how does it depend on concentration? How does the chemical potential depend on concentration? We haven't talked about that even with respect to G. Well, here's the equation. The chemical potential of some species A is given by the standard chemical potential that applies under defined standard state conditions, which in the case of a gas would be a pressure of one bar and a temperature of 298.16 degrees Kelvin. In the case of a solute in a solution, the standard state is an activity of one molar. We haven't said what activity is yet. So it's equal to this mu zero, which is the chemical potential for that pure compound in its standard state plus RT log of the activity of A. What's that? Well, to beg the question and not answer it, the activity of A is just equal to the activity coefficient of A times its concentration. Now I haven't told you what the activity coefficient is. Right? That's the activity coefficient. That's the concentration. So now we have a relationship between the chemical potential and the concentration of A. We all know what the concentration is and the concentration could be a pressure. Right? There'd still be an activity coefficient. All right? So if it's in the gas phase, it's a pressure. It's the partial pressure, of course. And if it's in the liquid phase, it'd be a concentration. And this activity coefficient, it turns out, can vary between zero and one. And we're going to talk more about that later on. But it's sort of a distraction for us now. So we're not going to talk about it right now. For our purposes, we're just going to equate for the time being the activity with the concentration. We're just going to pretend for the time being that this activity coefficient is always one. Okay? That's an assumption. It's an incorrect assumption most of the time. We'll talk about activity coefficients later. Okay? So we can relate the chemical potential to the concentration through an equation that looks just like this. Okay. Now, we already understand that G is minimized upon approach to equilibrium. We already talked about this. Equilibrium is at the bottom of this curve. But we didn't talk about why this is bow shaped. Because it's not intuitive that it would be. I mean, if this is the Gibbs energy of the reactants, this is the Gibbs energy of the products, why doesn't the Gibbs energy of every intermediate mixture of reactants and products fall in this dash line up here? It's the simplest solution to this problem, isn't it? And if that were the case, there would never be any equilibrium, would there? Because there would never be any curvature to this thing. No matter where you put this dot and where you put this dot, you're always going to have a straight line and it's never going to have a minimum where you would expect to find equilibrium, is there? All right. So the fact that there's a bow to this curve is sort of essential to how chemical systems work. Chemical systems have equilibrium. We need to understand that. This is not what happens. Okay. So let's do a simple calculation. I've got O2 on the left side, N2 on the right side. Half a mole of each. Now I'm going to open the valve. This reaction coordinate is the mole fraction of N2. When I open the valve, I'm going to have 0.25 moles of O2 and 0.25 moles of N2 after everything comes to equilibrium. Okay? And so I'm, the mole fraction is simple. The moles of N2 is 0.25 over the total number of moles is 0.5 and so 0.25 over 0.5 is 0.5 and so we're right here on this diagram now. All right? We're right here after all the mixing has happened. Okay? So let's calculate G. Well, G is just equal to the number of moles of N2 times the chemical potential of N2 and the number of moles of H2, except H2 is not in there. That should be O2 times the chemical potential of O2. It's all O2 and N2 here. Okay? And so what's the chemical potential of N2? Well, it's just given by the, because we're talking about a gas, that's the partial pressure of N2. This is the, this is one bar. We can just sort of ignore it. Your book usually does. Just call it one. Okay? And this is it, analogous term for O2, all right? We multiply by the number of moles. Yes, this is the chemical potential of N2 at one bar and 298.16 degrees Kelvin and that's one bar. Yes, your book just writes, it gets rid of the P0, just calls it one. Don't be confused by that. Okay? So since P is the same on both sides, we'll just call it P prime. At equilibrium, the total pressure on the two sides is the same. We'll just call it P prime. Okay? So after opening the valve, we're going to have P prime over 2 for nitrogen and P prime over 2 for oxygen because half the oxygen is going to go into the left valve, left bulb, half the oxygen is going to go into the, you know what I mean? Half the nitrogen is going to go, all right? So the pressure is going to go down by a factor of 2, right? Yes, this is all for oxygen, this is all for nitrogen. Okay? So the difference, the delta G final minus initial is just going to be P prime over 2 divided by P prime for nitrogen, same thing for oxygen and if I put the numbers in, half of them all, total of nitrogen, 8.31451, 298.16 Kelvin, all right? If there's a term for nitrogen, a term for oxygen, that's why there's two blocks here, all right? When I plug it into my calculator, I get minus 1,717 joules. Okay? So I can look at my plot then, all right? This delta is 1,717 joules when I'm at 0.5, that's not on the line. It's nowhere close to the line, the line's at 0, why is that? The line's at 0 because the mu 0 for nitrogen and the mu 0 for oxygen are both 0, turns out, look in the back of your book, there's a big table of standard free energies of formation, standard Gibbs energies of formation rather, you can look up nitrogen and oxygen, any other diatomic gas like that gives energy of formation is 0, all right? So we're measuring versus 0, okay? In other words, the curve has got to go through this point right here, it's got to go through that point and that point, so it's got to be a curve, all right? There's no way it can be a straight line between these two data points, we're just proving that. But the question is why? In other words, we know delta G is delta H minus T delta S, which of these two things is responsible for this curvature? Well, there's no change in the enthalpy during this reaction if these are idos, what is there? There's no change in the enthalpy if these are idogases, it's got to be in that term right there, it's got to be the entropy and of course when you think about it, as you mix these two gases together, right, and that you allow each of these two components to have more volume to occupy, the entropy of the system is going to go up, that's the driving force for this curvature that we're showing here, all right? And in fact if I plot the entropy, it looks like that, all right? Now what I've plotted here is actually T times delta S because I wanted to show you that it's exactly the exact positive reflection of this guy. If I put a mirror here, you get this, that's because I multiply by T, so that is going to be numerically equal to the depth of the delta G well, isn't it? It's going to be numerically equal. So entropy, I think this is moderately profound, all right? If you didn't have entropy acting in this process, there wouldn't be any equilibrium at all because if there's no curvature of this G versus progress of reaction curve, there is no equilibrium, all right? The universe would be fundamentally different than it is, all right? We have this curvature in delta G because of the entropy of mixing, that's what this is, the entropy of mixing. Okay, now this is one of my favorite molecules. It's called resveratrol. Why would it be one of my favorite molecules? Where is resveratrol found in our environment? One of the nice things about resveratrol is it makes things live longer and if you're my age, you're very interested in that even though it's only been tested in yeast, fruit flies, and worms. I'd like to think that human beings may also benefit from the same biochemistry that's going on with these guys, all right? Mice do well on resveratrol, all right? They don't get cancer as often when they eat a lot of it. Unfortunately, scientists feed them enormous fun, all right? And then they don't get cancer for God's sakes, all right? Here's the mouse without eating huge amounts of resveratrol, in acetone, mind you, happy day. The reason we humans like it is because it's found in red wine, California Pinot Noir's, oh my. All right, so this could mean that red wine is good for you, but it really cares. It's absolutely good. All right, why do we even mention it? Well, because there are two isomers, a resveratrol, and if you hit resveratrol with 350 nanometer, likely you get isomerization around that double bond right there. So there's a system transversion. And so this is like the simplest chemical reaction, all right? An isomerization reaction is like the world's simplest chemical reaction. Unimolecule, usually like or the stimulant. Isomerization reaction, see if we can figure out whether they can deliberately take the generic version of A goes to B, okay? Now, let's say that we're starting off with N sub A moles, that's why I'm calling it zero, because this is the starting point, and N sub B, this is zero, because that's the starting point. And if there's a change now, some reaction occurs, some of this A is going to get converted to B and some of this B. So if the reaction is going from left to right here, we're going to lose A and make B. And so this thing called the progress of the reaction or the extent of the reaction is going to tell us how far the reaction has proceeded from left to right, okay? So if that's minus point one, that's plus point one, and the units here are moles, the extent of the reaction has units of moles. It's not dimensionless, okay? And so if we start off with one mole of A and zero moles, B, we can convert this axis here into an axis in terms of the progress of the reaction, right? And if that's the case, this would be one and this would be zero, the progress of the reaction is zero when we got all A and it's one when we got all B, makes sense? One mole, right? The progress of the reaction always has units of moles. Okay, so we define this all for this plot of any value of the progress of the reaction as delta R of G. This is the notation you're both using. So this is the reaction gives free energy. So if I look at this point right here, all right, the reaction gives free energy is the partial derivative of the G function with respect to the extent of the reaction, right? It turns out you have to denominate everything in terms of the extent of the reaction in order for this to work properly, okay? So here, delta R of G is less than zero, yes, the slope is negative, here delta G of R is equal to zero, yes, the slope is zero, here delta G, delta R of G is greater than zero, yes, the slope is positive, okay? So we've got this thing called the reaction gives free energy. Except we're not supposed to use the word free. Okay, so delta R, that's meant to imply an incremental change. The conditions of constant pressure and temperature if the reaction advances by some small amount, all right? A will change by d moles of A, B will change by d moles of B, okay? Notice that the extent of the reaction is positive by definition, okay? So here, if we have a reaction from A to B, d and A would be a negative quantity, wouldn't it? Because A would be getting smaller. B would be getting bigger, so that would be positive, that would be negative, and so in order for this to be equal to this, I have to have a minus sign there because that's going to be positive by definition. Everyone see that? Okay, so these two things are equal to one another, and so if that's the case, I can factor out this, what is that, chi? Factor that out, and I've got this expression right here. And so in other words, this Gibbs reaction energy, right, is just going to be proportional to the difference in the chemical potentials. So delta G of R should be delta R of G, of course, is just the difference between the chemical potentials. It's just that, right? When I take DG, D chi, I get that guy. Okay, so it's obvious from this plot, but let's say it anyway, the Gibbs reaction energy changes continuously monotonically with the progress of the reaction or the extent of the reaction, therefore the composition of the reaction mixture. Yes, this is a smooth and continuous function. Okay, so that means that there's three types of reactions, reactions that are exergonic, exergonic, where delta R of G is less than zero, endergonic, where delta R of G is greater than zero, and equilibrium where it equals zero. And if you don't know what this word means, as always, Wikipedia can help you, means the releasing energy in the form of work, if it's an exergonic reaction, it can release energy in the form of work. You can do work, all right, if delta R of G is less than zero. All right, if you want delta R of G to be greater than zero, you've got to put work into the system to make that happen. Okay, but we still have not learned anything more about where equilibrium is located in terms of the extent of the reactions. To get there, let's assume A and B are ideal gases, then you will recall from lecture 17, isn't this lecture 17? Let's assume A and B are ideal gases, then you'll recall from lecture 16, yes, that's right, right? So the difference in the free Gibbs energy between some initial state and some final state is just given by this integral, where that's the molar volume, and we can convert that, we know if we're talking about an ideal gas, plug in chug, here's the expression as a function of pressure, okay, and we can define a standard molar Gibbs free energy as one that applies at a pressure of one bar and 298.16 degrees Kelvin, that's what that is right there. Okay, and so if A and B are gases, we can write their chemical potentials in this form. For example, because this is just the chemical potential of a single species, and it's going to follow exactly this form, I can write this equation right here by analogy to that one, and this is the standard chemical potential which once again applies at one bar and 298.16 degrees Kelvin, okay? So the Gibbs reaction energy, we said, is the difference between these two chemical potentials, and so if I write an equation like this for A and another one for B, I can take the difference, there's the B one, there's the A one, and when I take the difference, yes, I already said that. I've got mu B zero minus mu A zero, that's that, all right? And then the P zero's are going to cancel, I'm going to end up with P B over P A, and this is the standard now, Gibbs reaction energy. It applies at one bar and 298.16 degrees Kelvin, okay? And we have a name for this, right? This quotient between the partial pressure, B and the partial pressure of A is called the reaction quotient, something that you heard about in general chemistry, low of those many years ago, all right? We call it Q, the reaction quotient. Okay, so for every value of the extent of the reaction across the horizontal axis of our G versus the extent of the reaction diagram, we can calculate the Gibbs reaction energy from the Gibbs reaction energy to the product reaction species present at that extent of the reaction, yes. And so finally, at equilibrium, the reaction quotient is given by, so when this is zero at the bottom of that well, right, where you've got a horizontal line tangent to the curve, we can call this K, right? The reaction quotient isn't Q anymore, here it's Q, all right? Under the special case where that's zero, we can call the reaction quotient K, K or K equilibrium, all right, and these are then equilibrium values for these two partial pressures. And from this, I can just move this guy to the left-hand side and I've got minus RT log K. So there's a relationship between the free energy of the reaction, the standard free energy of the reaction, standard because this is a zero, and the equilibrium constant. For example, let's say that the delta G, the reaction is zero, right? If delta G is zero, Y can solve for K, delta G is zero, exponential of zero is one, yes, all right? In my extent of the E, in my equilibrium constant for the reaction is one, all right, which is going to be correspond to equal partial pressures of B and A, right, because the equilibrium constant is the partial pressure of B divided by the partial pressure of A, okay? Here's the reaction, the standard reaction free energy difference between A and B. It's zero, all right, the equilibrium constant's right in the middle. This is going to be a nice symmetrical curve, isn't it? Yes. And on the other hand, if this standard Gibbs reaction energy is less than zero, in other words, the final Gibbs energy minus the initial Gibbs energy is as shown here, all right, this guy is less than zero, now the curve is going to look like that. Now equilibrium's got to be over here, all right, and that means that it's going to be greater than one. It has to be greater than one. What does that mean for the equilibrium constant? Well, if we look at the equilibrium constant, here's the equilibrium constant refresher for a generic reaction, A moles of A plus B moles of B goes to C moles to C and D moles of D. We can write the equilibrium constant terms of the equilibrium activities of each reactant and product species, yes, it's just the activity of C taken to the C power times the activity of D taken to the D power and so on, products over reactants, right? We can write this in terms of activity coefficients and concentrations, yes, these are activities and these are concentrations and these are activity coefficients. And so what this means is that you've got a product rich mixture. You've got more product than you do reactant because you're on the right-hand side of one. And if the opposite is true, if you're going uphill from A to B, you're going to end up with an equilibrium constant that has the opposite. It's reactant rich. All right, we'll come back to this on Monday. Have a good weekend.


Now we'll go into the Q&A, we'll answer all your questions, the best of my abilities, especially all the stuff we just went over. And we'll go from there. But if you got to go, thanks so much for stopping by, hit the like and subscribe on your way out. That's it. Thanks for stopping by. Let's get into Q&A. Oh, yeah, one more thing. I just complete the videos for Cosmos or Adam and the near protocol staking on the ledger. Those will all be out on Friday as I'll be traveling to Houston. So that is it. Let's get into the good stuff. This is my favorite part of the day. Favorite part of the video. Oh, thank you, Vicky. Disclaimer, not financially, your tax advice, a chance rent in Brazil is just an opinion. Dan is not an expert, obviously, or a financial planner from your own research. Thank you, Vicky. With some people actually remind me to do that. I keep forgetting to do that. I should probably do that. Cam's videos, Voyager news coming out this week. FTX and the buyout. And what that means for VGX holders. Bitcoin without electricity is nothing. Well, that's no true statement has been. That's true. Why? Thank you, Sass Crypto. I'm looking dapper today because it's cold. Like I'm not good for cold weather. Texas is usually like a hot place. And right now it's around 5960 or something like that. Anything below 70 for me is no good. Before anybody talks about, ah, that's weak. You know, I can I live in a place where it's 5040 degrees all the time. Well, okay. Once you come and work construction with me, when it's 120 degrees or higher, see how we how you fare then because that's what it is here. All right. Tommy, sweat coin was best thing because sweater vests. That's a good one. I do like that. Quants still rolling. Boy, for the I don't own quants. I never really got into it. But tip of the hat to everybody who's been investing in a quant. And I got to tell you that the circulating the the max and circling supply is so low. Should have got into it. Missed that one. I'll miss a lot more. Come on. It's a good question, Beardy. When is Jarkie? Jarkie is a great person. I wouldn't need to wouldn't need to apply for a loan of dirty pay for a membership. But he's cheap. Beardy's got all his crypto or all his funds tied up on EOS. You can't expect much from Beardy. Come on. Anybody on Walmart? No, McDonald's. Gotta gotta make sure I get that that new job. I want to be able to afford a new green screen. That'd be nice. So he says, Bitcoin is the fool's gold. We'll find out. Let's see. And that's it. I guess I answered all the questions. So it says semiconductors need silver and gold. Exactly. Well, it's gold for sure. China makes the chips. The US eats them. I think that's all. Well, great. Going straight from Mickey D, the system manager with the vest. No, no, I'm going for straight management. The bill. If I if I go for the bill gates, look, at least I would have his money. You're welcome, Becky. Now, I guess this one's going to be tight. Not only money, blocks average 10 minutes, but sometimes two can be found in a single minute. In case it's going to take more than an hour. It happens. There you go. I just have never heard of that. So that's it. Oh, let's see. Now I know it's time to go because we're getting the. Oh, thanks. Thanks, admins. Everybody wiped out that that spam very quickly. I appreciate you guys. Yeah, Reaper man says, Hey, just got here. It's all over. Here's here's the whole story in a nutshell. So 85 minutes for the Bitcoin, Bitcoin to a minute block. Sometimes that happens apparently every 33 days. Never heard of it, but just want to bring a number of attention. Not a big deal. Miners are selling a lot and their reserves or how much they hold in their accounts in their wallets is on par with how much they had in 2010. Meaning they're selling a little bit more. And then Charlie was Cardano Oracle. It went on the main net. And then we went over just what's going to happen in the next hopefully 15 years. Seems to be a good thing actually. Oh, here in the crypto and energy and microchips. And that's it. Yeah. What are you going to participate in the Cardano Summit in Switzerland? No, I will be in Puerto Rico at that point, whenever that is. So they want to catch that dating site address. That's funny. Nice job, wrenches. Yeah, exactly. It does pay off. See where your combat vest and I'm not wearing any of my armor stuff anymore. The miners aren't really dumping. Remember, miners have to keep the lights on. So this happens all the time. Mine has got to sell. People got to sell. Just how it is. And I think that's it. And lastly, 1,830. That's a good way to stop it. All right. It's cold out here. I'm going to go inside. So everybody, thanks for stopping by. I do appreciate it. Again, well, there's a couple of videos coming out. One I talked to Tiffany Wong and Aaron Bennett from Celsius. And they gave me a timeline and an update of what's going on with Celsius as it pertains to all the different business opportunities and things that are happening. So I will edit that and give it out today. And on Friday, two new staking videos on near and not polka dot, Adam, Adam or Cosmos. I got to tell you Adam and Cosmos looks like a pretty great project. Now again, I'm super biased because I've been I've been dollar cost averaging it for some time now. But make sense to me. And we'll go from there. But that's it. So thanks so much, everybody for stopping by. I appreciate it. Enjoy the day. Probably be a lot of sideways action. That's all. So I'll see you guys in the next one. Adios.


So here it is. I'm going to be talking about grades, cables, and cells. And I'll tell you what aspect of those words I actually mean. And I'm going to be talking about art and craft, and mathematics and computer science. So let's start with the art. So I'm going to be talking about a particular design element, which is often called not work, sometimes it's called Celtic not work, because it's often associated with medieval Irish and similar traditions, where you have this idea of depictions of strands going under and over each other. Like I said, this is often associated with medieval Ireland. It actually goes back much earlier. There's a Roman mosaic. I've seen, you know, Egyptian and things that are most similar. And probably as long as people have been drawing things, they've been doodling strands that go under and over each other. They're still doing it today. A couple of modern examples that I just sort of pulled off the web. All right. So personally, the study of this was inspired by knitting patterns. This same sort of under-and-over design in knitting is known as a cable. My wife Lana there may be able to tell you why. I don't know why. You'd be perfectly honest. But knitters often do these. And it's not done by actually making strands and tacking them down to the fabric, by the way. It's a little bit of an illusion. There really is a 3D effect, but it's not actually things smacking down right on top of each other. But it looks like it is. Just similar to where you draw things. It looks like things are passing over and under each other. You can also do it if you don't knit. I don't knit, but I do crochet. Theoretically, you can do it in crochet, although it's harder. I've tried it and don't come out anything like that. Nice. You can do a similar effect, even more of a trompe l'oeil and less of an actual strand going over and under. But you get this similar idea. Actually, for this particular design, you really maybe see it even better in the pattern. This is the pattern that somebody might be following as opposed to the actual. You sort of see that this one is coming along and it goes under and then over and under. It gives the impression anyway. This is a common design motif in several sorts of art. Of course, there's an actual weaving where you really do take either threads of fabric or in this case, I guess, a reed basket in a traditional Native American pattern. You actually do pass things over and under each other, so that's another example. This study of things like this mathematically from a mathematical point of view, the earliest I've been able to trace it back is Vandermonde, probably maybe known from the Vandermonde Determinant. Only route turns out four mathematical papers. One of them started knot theory, topology, and graph theory, well, maybe not quite started, but a really early paper in all three of those things. Here is an example of maybe what today we would call knot theory from that paper. Gauss was interested in looking at braids. He tried to work out this notation where complex numbers represented the crossings and tried to use the imaginary part to identify how many times it had gone under or over another strand. Never quite got it to this satisfaction like many things Gauss did, and therefore never ever am publishing it like many things Gauss did, but after his death they found this picture in his notebooks. Today, braids are studied by topologists and group theorists. Topologists, because, well, topology is the study of things that are the same if you sort of pull and push and stretch them around, but you don't break them. So you can sort of see how this strand here was pulled under this strand there and then over that strand there and this end there was pulled over a little bit there. Topologically speaking, these two braids are the same. So those are considered the same. When you're studying braids, the other thing you're not allowed to do is loop a strand over the top and behind another strand. So you can either just sort of think of the universe as ending here or maybe think of it as extending infinitely off of the power you prefer. I am going to be, and in fact you'll see in these diagrams nothing even loops back on itself. And from a knitting point of view, that makes sense because, well, if you might hold up your knitting for people to see, I just want to see what it looks like when you've got a needle and things dangling from it. I'm sure you've seen that before, but the idea is you're basically building it up row by row from the bottom to the top. And so you don't usually turn around and start knitting in the other direction from the top to the bottom. Likewise, I'm going to be thinking about strands moving from bottom to top, although if you ask the topologist they might disagree. But I said this was also studied by group theorists. Group theorists like to think about taking braids. If you know anything about group theory or even if you don't, the idea is just that you have a rule that says you combine two things and you get a third thing. Well, you can combine two braids by stacking them on top of each other. So if you think of stacking this one on top of that one, then you get a braid which starts here and it goes up for a while and then it goes under the next braid over. It goes up and goes under. And this one is going up there and then straight and this one is just going straight so you can see that with a little bit of topological pulling and pushing around, this braid is just the same as those two stacked on each other. Sometimes you get simplifications if one had gone over this way and then over the other way or the other braid and then it would just pull out straight. So you could get two braids. Well, by the end of the braid, you have inverse braids. It's related to the symmetric group which also, you know, you can think of this as interchanging these two and this is interchanging these two except if you interchange two things and then you interchange them in the same direction again, they don't cancel out. It matters what direction you're switching things or anything whereas to cancel out you have to switch it back in the opposite direction. So for those of you who really know or remember your algebra there's a homomorphism, a subjective homomorphism from this on to the symmetric group but with an algebra problem. It doesn't really matter. I'm not actually going to talk about that for the entire rest of the talk. In case there are any algebra acids in one that feel very special. I'm actually going to talk about something else entirely. I'm going to talk about cellular automata which is actually an idea used mostly by computer scientists. I don't know about mostly. Used a lot by computer scientists although it's a purely mathematical construct that's theoretical computer science. It's a very simple model of a computational system. I don't know if I want to call it a computer. It's got a finite number of cells in a regular grid. So you've got this regular grid of cells that extends for a while and then maybe stops. You have a finite number of states. Each cell can be in one of them. Just a finite number of states. So maybe this has got a dark gray state and a light gray state and a white state and that's it. Each cell is in maybe one of three different states. Each cell is a well-defined finite neighborhood and is only affected by cells in its neighborhood. So this is a picture of what's called the von Neumann neighborhood which means each cell is influenced by the cell directly north, south, east and west of it but not by the one's diagonal and not by the one's more than one away. But you can pick that neighborhood any way you want. You can pick it so it includes the diagonals. You can pick it so that it goes two away instead of one away. You can pick it so that it goes one away two away in some directions in whatever you want to pick. As long as it's the same for every cell as long as it's the same for every cell. Time is considered to move into discrete steps and you have an update rule that says that in each time you look at the neighborhood of a cell and use the states in the neighborhood to determine the next state that that cell is going to begin and every cell is going to use exactly the same rule to do that. So it's a very simple computational model but the interaction of maybe a large number of these things can get you interesting results. So that's the theory of it. So an example on here we've got a couple of experts on this in the audience. You're going to get the wrong. Okay, all right. Tell me if I'm doing anything wrong. This was invented by a mathematician named John Conway. Supposedly this was first played on a tile floor in a bathroom. This was first played on a tile floor in a bathroom. I've got lots of nice little squares to put markers on. The grid is two-dimensional and square. You don't even have to have a square grid although it's most common. You could have a hexagonal or a triangle. This one, the game of life, is played on a square grid. The cells are in two different states which are usually called live and dead although unoccupied and live might be more accurate. The neighborhood is the eight cells, so horizontal, vertical or diagonal, so all eight of those are going to influence the center cell. And the rule is that if you have a live cell and it is two or three live neighbors, so this cell is two live neighbors, that cell is three, that cell is three, those cells are going to stay live. If you have a live cell with fewer than two live neighbors it dies of loneliness. Very, very sad. And if you have one with more than three live neighbors it dies of overcrowding. On the other hand, I think those cells just disappear. There aren't any dead bodies floating around to really get you down. That's good. A dead cell that has exactly three live neighbors, so these two cells have exactly three live neighbors, so I said dead maybe you should really think of as unoccupied becomes a live cell, so three parents give birth to a new cell. I don't think this was inspired by a particular biological model. I think that they just experimented with the rules for a while until they got something interesting. Any other dead cell if it's not surrounded by exactly three live neighbors and this one has two, that one has four, those cells just stay dead or stay alive. Seems like a very simple sort of thing, but it actually creates some very interesting and complex patterns. So this one is a pulsar. If you start like this it evolves like that and gets back to where it starts after a couple so imagine some really interesting animations. So this one just cycles but there are some that speed off projectiles in directions. There are some that just truck across the I want to say screen when you're done with a computer animation across the board I guess you should maybe say if it's a game. So lots of really interesting behavior can be done like this. So that's a one of the really earliest examples of a cellular automata. There's another well-known class of cellular automata. They're called elementary cellular automata. These are about as simple as you can get. These are even simpler than the game of life. They were popularized by Stephen Wolfram who was the guy who invented Mathematica and he wrote this book The New Kind of Science about how he thought that every piece of science in mathematics should be modeled with cellular automata and that should really be the new way of doing science. That seems a little bit extreme to me but you can do lots of interesting things even just with these simple ones where you have a one dimensional grid so instead of a square it's just a line of cells two states white and black or sometimes 0 and 1. Sometimes I'll say 0 and 1 but I really need white and black. The neighborhood is just three cells the cell you're talking about and one cell on each side. So how do you specify a rule? Well, you've got three cells in the neighborhood. Each one can be in one of two states so how many different possible neighborhoods have we got? Each of which is in one of two states 8, right? 2 to the third So for each of those 2 to the third possible neighborhoods you have to decide is the center cell going to be 0 or 1, white or black? So in this case someone has sort of arbitrarily decided that if there are three black it's going to be white other than 0 and in this case it's going to be white 0, 0, 1, 1, 1, 1 and 0 Ok, so is that completely determines a rule?  I asked this by the way at an Illinois MAA section meeting last week and none of the professors were getting it right I think the undergrad students Yeah, and now that I've said that does anybody actually want to try it? So you've got 8 different possibilities for the neighborhood in each case you have to decide whether it's going to be a white or a black So We're trying to work out the details of my head I was thinking about it as a binary number So you could think about it as an 8 bit binary number in which case how many are there? 64 I don't know 356 doing 8, 256 And this one and Stephen Wolfram just decided to take that 0, 0, 1, 1, 1, 1, 1, 1, 0 and make a binary number out of it and so this is rule 30 because that's the binary number for 30 so that just gave him a nice way of numbering all of those rules from 0 to 250 Rule 90 is a picked out some of the most interesting of those rules So rule 90 is a nice interesting rule Now, Stephen Wolfram is obviously not a knitter because time in his diagrams is starting at the top and progressing down to the bottom so I'm going to reverse that in a bit but I did copy this from one of his diagrams which was plus a math world so if you start out with just a single black square then the next time you step it all like that like that and starts developing out the Serpinski triangle fractal and the longer you let it run if you have an infinitely wide board the longer you let it run the more of the triangle fractal you get This is also a really nice rule because it's an additive rule which means a couple of things but if you take one of these patterns and you smack another pattern that's made by a different cell smack it down somewhere else and then say white plus white is white and black plus black is white so 0 plus 0 is 0 and 0 plus 1 is 1 and 1 plus 1 is 0 addition modulo 2 or XOR if there are any people who prefer computer notation you get the pattern that you would get if you just started out with two black squares to begin with so it's really convenient to be able to predict what's going on because you can build larger patterns out of smaller ones just by using this addition that makes it theoretically very nice to work with however Wolfram was particularly interested in much more complicated behavior something that appeared to be chaotic or pseudo-random or something like that so he was looking at a different rule well if he was looking at this rule 30 and he was looking at what happens if you get a rule 30 and he noticed that it has a well like I said it kind of seems like it behaves chaotically although it's hard to define what catech actually means for a discrete system and I'm not going to try but it's got a pattern going down this way and kind of a pattern going down that way and somewhere in the middle it transitions to something where it becomes much harder to predict what's going on he didn't get very far in proving this to be perfectly honest he couldn't even quite prove this conjecture that if you draw a line right down the center it never repeats alright well the rule 90 never repeats either per se but it's very predictable and this is not very predictable not clear what's going on there exactly and so one of his I'm not sure students, protégés proved at least that at most one cell is periodic so almost all of them maybe none of them all of these produce a periodic sequences and and also any two adjacent cells definitely produce something a periodic which I guess is a simple formula of any and so Wolfram was confident enough that this was an unpredictable sequence that he used to do this as soon as he ran a number generator but you remember you started with a very simple rule and just one square and you're getting something very complex coming out of it so another example of that is in rule 110 which was proved to be computationally universal in the sense that it can be used to simulate any Turing machine you don't know what a Turing machine is again it's another one of these mathematical abstractions of a computer but this is a very rich one it is widely, maybe wildly also it's widely believed that anything that you can compute with any algorithm you can compute with a Turing machine whether this is actually provable or not depends on how you want to define an algorithm some people just kind of define an algorithm by anything you can compute by a Turing machine in case it's proved I guess but people think many people think that computation is completely captured by the idea of a Turing machine and therefore computation is completely captured by one particular cellular automaton rule 110 and if there are any complexity geeks in the audience it can be done with only polynomial expansion which is important for people who feel that that's important I don't know what else to say on the single cell rule 110 doesn't do anything especially interesting because if you just have a one bit input you can't do anything especially interesting with it but the idea is that you can use rule 110 to simulate a cyclic tag system I'll explain what that is in a bit and then you can show that a Turing machine can be simulated by a cyclic tag system I think that the easiest thing to do which I'm not going to do is to show that a cyclic tag system can simulate a non-cyclic or not necessarily a cyclic tag system once I tell you what a cyclic tag system is though I think that you'll believe that that can simulate a Turing machine the interesting part is that 110 can simulate a cyclic tag system a cyclic tag system has a data string and a list of rules which is really just a computer program except here's where the cyclic part comes in it's a computer program that just consists of one big loop that's all it is so it's a fairly restricted computer program and there's only one sort of operation but it has a branch in it so at least you can do something with it if the first data symbol in your string is a 1 then you tack your rule onto the end of your string if the first data symbol is a 0 then you don't do anything and then you delete the first data symbol and repeat so if you have these three production rules you just go 1, 2, 3, 1, 2, 3, 1 and so on and you start with this data string then you look and say oh, okay, the first thing in the data string is a 1 so I'm going to stick this rule onto the end and delete the 1 okay, got it 1 still stick the next rule onto the end and delete the 1 now I'm going to 0 so don't put anything on the end, delete the 0 don't put anything on the end, delete the 0 well now you're back to this 0, 0, 0 rule and you've got a 1 and you put the 0s on the end and now you're back here so you go use it and you can see that as it goes on it's going to be doing different things with the same rules depending on whether it's got 0s or 1s in front so I hope you can believe that you could maybe write a computer program in this way it would be difficult but it has branches, it has loops so it has the basic things that you need now here's how you simulate that with the cellular automata you need a representation of the data string and that's these vertical lines here I think that the gray thing is a 1 and the black things are 0s I'm not positive about that and you need the production rules so that would be these things coming along this way and then you need clock pulses which come along that way so each interaction clock pulses is a time step so in each time step you're going to have the data string hitting some production rules and it's going to chug along a bit and then it's going to rewrite its thing and then move along to the next time step and I'm going to go through all the gory details but this is I think one time, actually it's not even a whole time step the clock pulses aren't even on here this is all between two clock pulses but you can see some data coming here hitting some representation rules and doing some interesting things and some different data strings coming out the other side so here's a data string that comes out the other side and here's one that doesn't so I guess that's maybe a 1 and a 1 changing into a 0 I don't remember exactly but there's a lot of expansion you need a very large piece of things in order to simulate each clock step but it is according to the theorem only polynomial so for a computer scientist or a theoretical computer scientist or a mathematician that's good enough wouldn't want to do it in an actual practice so that's cellular automata please feel free to interrupt with any questions as we go along so how do I get started in combining these two things these two ideas of braids or knots or automata well two things happen and I was trying to reconstruct I'm not absolutely sure what order they happen in anymore but what is it that I was talking to a fellow named Jake Wildstrom who's a mathematician who likes to crochet sort of pins and trindle fractals like this one here and he said somebody had asked him to write up a pattern for crocheters of this and he was complaining to me that you shouldn't really need a pattern because after all it's just produced by the simple rule if you have a raised bit then you make an unraised bit and if you have two unraised bits then you make two raised bits you make an unraised bit and if you have one raised bit you make a raised bit I agreed that you really shouldn't have to write out the whole pattern for that but probably people who crochet are more likely to want the whole pattern spelled out and not just that simple production rule but I stored that in the back of my head somewhere and the other thing is that I was looking at something that Lana was knitting with cables on it and I noticed that she was doing it without a pattern and it was a fairly long and intricate much more intricate than this one here a fairly long intricate pattern and she appeared to have it memorised and I remarked on that and she said, well you don't really have to memorise the whole thing because there's a lot of parts where you just remember that if you're going this way you do that and if you're going that way you do this and if you're crossing then you do that and so I sort of sketched that way in my head and I got to thinking that sounds like the same thing we were talking about here and this right now is all about cellular automata so maybe these braid cable patterns also and I did a little bit of research found that somebody had had this idea before this idea wasn't original to me a woman named Debbie New had written a chapter of one of her books where she explained about this but she only really explained how to do things like this pattern where basically they all go straight except sometimes they cross and the cellular automata rule just determined cross or no cross didn't determine which one crossed which direction or whether it went off in some other you know just crossed so I said well okay that's a start but I think we could probably work more into it than that so in my model each still is storing four bits of information which I don't want to ask you to answer this one ordinarily that would be 16 different states but we're going to sometimes ignore some of the bits of information so the first two bits of information is there a left strand or not yes no yes no is there a right strand or not no no yes yes okay that's easy if there are strands are they straight or slanted that's another bit of information except sometimes right if there are no strands it doesn't matter whether they're straight or slanted so in that case we'll ignore that bit of information but otherwise it could be straight or slanted straight or slanted but then if you have two strands and they're slanted well then you can think about did they cross this way or did they cross that way and so in that one situation you're going to want one more bit of information telling which way the process goes so there's eight different types of cells but there's four bits of information some of which are ignored in some situations what about the neighborhood it seemed convenient to have these be a brick wall neighborhood again like knitting patterns time is moving from bottom to top here and each cell is going to be influenced by these staggered cells below left and below right except I mean in a sense these aren't below in a sense these are before just remember the second dimension is the same as the time dimension so if you have eight cells theoretically and two neighborhoods theoretically there could be eight times eight different types of neighborhoods so you could have 64 and theoretically each of those could have eight different results and that would be a really really really large rule set so I decided to put some more conditions both to make it a little better physical model and also to make it a little more tractable so the first two parts I broke down the rule system into four simpler rule systems and the first two parts just say strands continue they don't suddenly disappear they don't suddenly split into two they don't suddenly appear out of nowhere in knitting you can sometimes break these rules but I thought just for starters let's set out these rules so if there are no no strands then there continue to be no strands if there's a strand on the right below and on the left below then there are two strands above if there's a strand on the right below if there's a strand on the left above because of brick wall standard and so on it doesn't matter for these purposes whether they're straight like this or slanted like that this is just saying that strands continue they don't break or appear out of nowhere or loop around back on themselves or anything like that third stage a little more interesting controls whether the strands are operator diagonal and so I decided that maybe the important question was just whether the strands below were upright or diagonal or missing decided to make it a little more interesting to have a separate category for missing so the strand on the left here could be diagonal slanted it could be missing, it could be upright and likewise the one on the right could be diagonal and you're missing, it could be upright so again that gives me nine possibilities I put the middle one in for symmetry although if there are no strands then there are no strands and it doesn't matter whether they're upright or not but just for symmetry I left it there and so the zero says that a strand is going to be straight up here and the one says that it's going to be slanted and if there's planning they might cross but this one doesn't affect the crossing at all crossing is controlled by another one so in this case I decide whether the cell below is a cross with left over right or just a single strand or a cross with right over left and oh I meant to add this do I have another? No, I have another I decided that anything uncrossed regardless of whether it was one strand or two strands as long as it didn't cross it was going to be in this row here so that gives a little bit of interaction between the two because something which if it's diagonal it's crossing and would therefore be in one of these if it's not diagonal it's not crossing despite the fact that there are still two strands there and so it might be in this row here gives a little bit of added interaction provides a little bit more interest okay then I have to decide I've got the neighborhood I've got the the types of cells a little bit over the fact I said there was a finite grid but then I didn't tell you what happened at the end of the grid and actually Conway I don't know what John kind of originally did in the game of life Steven Wolfram just decided to make the grid infinite so that he didn't have to worry about the edges you could have a special kind of state for edge cells that things maybe got absorbed into or bounced off of if you're knitting a sweater you often see cables on that sweater can you stand up you might see does the cable ties you to your arm around to the back or does it get absorbed into the side seam you maybe haven't even left to see that one's coming another common thing to do is to have the cable sort of absorbed into the side seam when you get to the side seam or maybe bounce off the side seam but then you have to decide exactly how the bounce works so I took the easy way out and just made the grid sort of just wrapped around the cellular automata people call this periodic boundary conditions although what they really should say is there are no boundary conditions because they're a boundary but that's what they call it and that seemed to be the easiest thing so I'm going to stick to socks and hats and very simple sweaters these guys are solidical right I can get some very nice patterns I can produce patterns like the Serpinski Triangle Fractal this is a bit of a cheat remember because there aren't really two triangles here there should be but in fact these are the two halves of the same triangle because it's wrapping around but of course if I just made it larger I could get as many triangles as I wanted I just have to decide how many I want to make sure it's that large weaving patterns so very typical so this is rule 68 for the thing that determines whether it crosses or whether it slants or doesn't slant and which I call the turning roll sometimes and rolls 0 for the crossing roll which just says all the crossings go the same direction so very simple but I can reproduce some of these practical patterns these in this case nothing ever turns it just keeps going in the same direction that it's going so the turning roll isn't active but the crossing rolls decide when it's crossing which one is crossing over the other so this is a nice classic basket weave pattern this is maybe not something that you would want to weave a basket out of but it's a graphic design element there's nothing wrong with that this is a traditional three-strand braid when you're back when you're at summer camp and you're with friends or whatever friendship bracelets I don't know just a very standard three-strand braid a four-strand braid very common they are in actual they are in my model with a this is the same two sets of rules actually the only difference between these is if you start with this you get a three-strand braid and you start that same rule start with this wider pattern as you want and that's how the braid is again giving the idea that we're actually capturing something interesting with these rules that this is the braid rule the traditional braid rule crossing I was looking for pictures online of these different braids think I've seen this braid before I've never seen this braid before some people have has sort of a large jump where you go over I guess you go over three strands of that under one and then under three and then over three and then under three when you actually pull it tight gives you a three-dimensional very round sort of braid because these long loops pull up into the this one I guess pulls up into the front and the next one pulls back into the back so that's nice and I was satisfied to see that I could model that with my rules if you had various other sorts of interesting I hope for interesting decorative cable patterns these are starting to look more like something than you might actually mint and in fact I want to like the one on the left so uh oh that's okay so yeah so yeah that's pretty good the lighting is pretty good to show off the relief here so you can see that this is passing over and under and over and I need to know that for a while so there are some cable patterns there technically only one of those socks was rule 47 the other one is the mirror image and every so often that asks me if I've what rule produces the mirror image and I still haven't remembered to sit down and work it out it's like you have to flip all the bits and then flip the high end with the low end or something like that I worked it out once and I immediately forgot alright so what do we know from a point of view of theorems about this well if you don't have to worry about missing strands and you're only using a crossing or a turning and not the other one then basically this is an elementary cellular automata where you ignore the middle cell of the neighborhood so this is exactly the same as Wolfram's rule 90 with no all the crossings in the same direction just determining whether you cross or whether you turn make one of them white and one of them black and you'll get exactly the same pattern for exactly the same reason so okay that's something we know I guess somebody asked me once about the chaotic or chaotic like a periodic behavior well the since the width here is finite there technically isn't any well it's not technically there isn't any a periodic behavior since there's only a finite number of states the system total can be in eventually it has to repeat but you can ask how long that repeat is so first thing I worked out was just not having the weaving patterns where nothing ever turns everything just goes straight only the crossing rule is active for a given width M I worked out how long a repeat can possibly be so the idea here is that there's a natural repeat if M is the width there's a natural repeat of 2M after which things sort of potentially get back to where they started if you've got a missing crossing strand here the missing crossing strand rotates through and after 2M it gets back to where it started so that's a natural sort of repeat then you just for the ones that do cross which one is on top is all you have to worry about since nothing is turning and I tried to make the crossings in one direction red and in the other direction blue I don't know if you can read that out there if there are all M strands present to all M crossings then you simply say okay there's 2 to the M possible arrangements of the crossings but then the 2M sort of gets lost because there isn't that missing strand percolating through there is a shift from this stagger to that stagger so the maximum repeat is the least common multiple of something that goes up to 2 to the M and something that goes up to 2 okay on the other hand there's only M-1 crossings if you leave one out then there's only 2 to the M-1 possible arrangements but 2M different shifts so now you've got the LCM of something less than or equal to 2 to the M-1 and something less than or equal to 2M and you can work with there are fewer crossings than that it has to be even smaller you can work it out fairly easily and then you put that together and you've got about M times 2M but you can work out the things where the least common multiple doesn't really get you anything and this is as long as the repeat can be except for trivial M's I haven't gone anywhere near that I'm not I don't expect that you can't get very close to that really in practice but I haven't proved it if you have a given width how long a repeat can you find if you look at all the possible rules and all the possible starting positions well for a given width you can find one if M is greater than or equal to 2K you can find one that's at least the LCM of 2 to the K plus 1 and 2M rows long and basically I just found it I did a little bit of a search and after a while I realized that crossing rule 100 is an additive rule I'm going to talk about how additive rules were easy to analyze theoretically and so I worked out that for that rule I got a nice long repeat of 2 to the K plus 1 just in the crossing what's really going on here I finally figured out after looking at a lot of examples is that what you've really got is you've really got another one of these elementary cellular automata trekking along but sort of slanted that way but with a boundary so this is actually you can look at this as an elementary automata that doesn't wrap around and so eventually I found something very like this result in the literature which said that the repeat for the pattern that goes diagonally is at most 2 to the K plus 1 it is 2 to the K plus 1 excuse me if M is larger than 2 to the K you work out the equal part separately and then you have also this cyclic shift which gives you the 2M I thought for some time that this was the best you can do it's not for M up to 5 I ran maple simulations because maple is what I had handy and then I ran out of patience and decided that that was probably good enough but then again I was reading some of these results on an elementary cellular automata and I realized that for the elementary cellular automata with periodic boundary condition and certain prime numbers and additive rules you can get a really long repeat not quite as long as my bound here but for certain cases you do get a nice long repeat and for M equals 23 you actually get longer than this my current guess is that additive rules and either M or M minus 1 crossings are going to give you the longest repeats but I haven't proved that and I haven't shown which situations it's going to be M in which situations it's going to be M minus 1 it's a lot of work left to do if anybody wants a project either for themselves or for some students interested in cellular automata there's a lot of great stuff there and this is just if you only worry about crossings and things don't turn I've got even less as things do turn if there's only one or two else there's only one strand you don't have to worry about crossings at all so that's easy if there's two strands then you don't have to worry very much about crossings and again it's not too hard just to work out but the maximum repeat there's the natural 2M shift and then there's you can do this little hop here which gives you another factor of 2M plus 1 and the LCM of 2M and 2M plus 1 of course is those relative to the prime you can tell what's doing with only one or two strands but that's not especially interesting more than that you really don't know and like I mentioned before both rules are active I set it up so you have two elementary cellular automata going but one of them sometimes interferes with the other in the case where they don't cross then it bleeds from one into the other but there's a really nice probably accessible structure there that I have not just really started working on yet lots more of the situations for the length of the maximum repeat to work on computational complexity so the maximum repeat stuff is sort of related to the air periodic behavior this is related to the computational universality of rule 110 again on a finite grade you can't get true computational universality but you can say how hard is it to answer a question like is this cell eventually going to be in this state it might be again if there are any complexity geeks it might be as hard as p-space completeness to figure that out but I don't know if it is or not different edge conditions I really did at some point want to go back to the break groups and say which elements of break groups can actually represent with these things again because of the finite limitation there's a limit it can't be all of them but which ones reversibility there's an idea of reversibility of cellular atomic rules which rules can you go backwards and uniquely trace back through a pattern given an ending condition I don't know and then lots of people have suggested well what if it does bounce around and turn back on itself you need a two dimensional grid to represent that and time would be a third dimension you could represent maybe by color or by some sculptural representation in three dimensions which would be really cool but I don't know what to do with it yet so lots of future work to do with that well thanks very much thanks a lot to my wife for lots of encouragement any questions for our speaker I'm just wondering if you have mentioned some rules are added and some must not be and those have been delineated is that an easy question to ask so for the most elementary situation you ignore the one in the middle like I am because it's a brick wall and then there are only 16 really different rules and it's fairly easy to delineate them for all 256 elementary rules you can say which ones are most of them are not added and you can say a little bit about how nonlinear they are I have found there seem to be special cases where people have studied threshold rules where the more cells you have the more likely it is to turn the next one on the more cells that are ones make it more likely to have the next ones and and several other specific types of rules have been studied well from the start of the grant program we categorized the complete behavior of alternate 52 rules what about the original problem of representing knitting structures with the rules and are the rules helpful to a winner like can you knit by rule when you had when you had rule 47 did you actually knit those socks you actually knit those socks from a real pattern didn't you I knitted them from your diagram that you had generated but I had to figure out the specifics for how I was going to interpret each of the cells in knitting after I had knitted I didn't stare at the rule what was going to happen but after knitting for a bit whenever this one moves this way then it's going to go straight first so I was using the rule but I didn't start that way I think the answer is some of each I could have learned the rule first and then surprised myself then so far I think the most the thing about this that would be most interesting to a knitter you can correct me if I'm wrong is that I think it's a really neat way of generating patterns that somebody might not have thought of I don't know if anybody would have ever sat down and designed one of these this one maybe it has a certain erogenous to it that doesn't spring easily to the mind but is still appealing thank you again