This lecture is part of a series of lectures on elliptic functions and in today's lecture I'm going to be talking about the functions Studied by Jacobi which are usually denoted by SN, CN and DN And these functions are kind of a bit out of fashion. I mean books on elliptic functions in the 19th century were all about Jacobi's functions, whereas They seem to be almost unused in pure math these days. I mean Lang wrote a quite large book on elliptic functions Last century and doesn't even mention the Jacobi functions So I'm not going to go into too much detail about Jacobi functions Instead I'm going to use some mainly as an excuse to discuss line bundles over an elliptic curve And so I'll just quickly recall Basics of elliptic functions. We have a lattice L In the complex numbers and our elliptic function is periodic in L So f of z plus lambda is equal to f of z whenever lambda is in L And as we saw in previous lectures f must have some poles And the sum of the residues of the poles of f That's in a fundamental domain of course must be equal to zero As you can see you remember we just integrate f The z round the boundary of a fundamental domain and this is zero because it cancels by periodicity So if we want to find the simplest sort of functions We might try having one pole But one pole doesn't work because if we had just one pole of order one Then the residue would have to be zero so it wouldn't exist So there are two possibilities. We can either one pole of order two And residue zero or we can have two poles of order one where one has residue minus the other And roughly speaking this is the approach taken by Weierstrice Which we discussed in the previous lecture and this is the approach taken by Jacoby Which we're going to discuss today And Weierstrice's approach is easier for the following reason Suppose we've got one pole in a fundamental domain where do we put it? Well it's pretty obvious where to put it The pole is going to be at the point z equals zero Or more generally for z in any lattice and any other choice to just be stupid But if we've got two poles where are we going to put them? Well first of all there's a constraint because some of the poles If there are poles at z1 and z2 then z1 plus z2 Must be in the lattice as you remember from lecture two So inside a fundamental domain Here we've got a fundamental domain We've got to choose two different points Whose sum is in the lattice and if you think about it there's no really good chronicle way of doing this z1 or z2 can't be zero because then the other would have to be zero without a double pole at zero And similarly you see that the other obvious choice would be these half lattice points And these don't work either because again z1 would have to be equal to z2 So we've got to somehow choose two points That sum up to zero inside this fundamental domain As you can see there are lots of ways of doing this And that doesn't seem to be any way that's picked out for being rather special And this is one of the main reasons why Jacobi's approach to elliptic functions is a bit of a mess compared to Weierstrich's approach Because you've kind of got to break the symmetry of your problem by choosing a pole somewhere What Jacobi did in fact was he chose two points of order four like that But there are lots of other ways of doing that I mean you could choose these two points here or you could choose these two points here and so on So you've got to break symmetry And this breaking of symmetry means that everything gets more complicated Because you've got to write out everything several times instead of just once because you've lost the symmetry Well there's actually a way of dealing with the Jacobi functions that doesn't avoid breaking the symmetry Or at least not quite so much So there's a solution We can have just one pole of order one in the fundamental domain Well I just said you can't have just one pole of order one if you've got a periodic function So you have to modify the periodicity slightly So the function is going to be not quite periodic And for an example of this let's look at the function sin of z So sin of z plus 2 pi is equal to sin of z So it's got a period 2 pi But instead we could say sin of z plus pi is equal to minus sin of z So what we have is we have here a fudge factor And the function has period pi Except it's not quite a period it's periodic up to this fudge factor And we can do the same or doubly periodic functions So what we want is f of z plus lambda is going to be fz times c lambda Where c lambda is going to be some element of complex numbers And now we want f of z plus lambda plus mu should be equal to f of z of mu plus lambda Obviously which gives you c lambda times c mu is equal to I guess this should also be f of lambda z plus lambda plus mu So this gives us c lambda c mu equals c mu c lambda equals c lambda plus mu So c lambda is actually homomorphism from the lattice L To the non-zero complex numbers And we're going to look at the case when c lambda is equal to plus or minus 1 So this is the simplest case and obviously c is determined by its values on omega 1 and omega 2 So there are four possibilities We could have c omega 1 and c omega 2 could be plus 1, plus 1, plus 1, minus 1, minus 1, plus 1, minus 1, minus 1 Now this case is where the functions are periodic and we've already covered that But you see that there are now three more cases And these are going to correspond to the three Jacobi functions And now you notice if a function is not quite periodic in this way Then it still has periods but the periods are going to be slightly bigger than omega 2 So here the periods are specified by omega 1 and omega 2 in this case But here you see omega 2 changes the function by minus 1 So if you have 2 omega 2 then the function is going to be periodic And similarly this is periodic with respect to 2 omega 1 and omega 2 And this is periodic with respect to 2 omega 1 and omega 1 plus omega 2 So here these are the three index 2 sub lattices of L And one of the annoying things about Jacobi functions is the lattice they're periodic under keeps changing I mean so SN is periodic under some lattice and CN is periodic under some different lattice And this is very annoying But if you don't think of them as being periodic functions But just functions that are periodic up to sine then the lattice stays the same each time So this makes it a little bit easier to think about them You can ask why is C lambda equal to plus or minus 1 And the answer is there's no good reason We can take C lambda is an nth root of unity for some n And then we get n squared possibilities for the different sorts of functions And this means that instead of getting you know when n was 2 we got 2 squared possibilities One of which is periodic so all together we would get n squared minus 1 elliptic functions So Jacobi was doing the case n equals 2 of this and and you can do other other values of n But it just gets a bit messy I mean you would have you know if n is 3 you would get 8 elliptic functions instead of just 3 Actually when I say there's no good reason that's not quite true There is actually a reason why n equals 2 is rather better than the other cases that n equals 2 corresponds to something called the symmetric line bundles The line bundles that are isomorphic to their dual and that does make things a little bit easier Well let's take a quick look at some properties of these not quite periodic functions So we're going to have f of z plus omega i is equal to ci times f of z So you recall for periodic functions we have the number of poles is equal to the number of zeros And this still holds for this case here and you can prove it in much the same way you remember We can count the number of poles minus the number of zeros by doing this funny integral here as in complex analysis And we take the integral around the boundary of fundamental domain like that The bits on the top and the bottom cancel out and the bits on the left and the right cancel out so that's zero A slightly more interesting case was for periodic functions we had the sum of the zeros minus the sum of the poles And for periodic functions you remember this was in the lattice L And the way we got it was by looking at the integral 1 over 2 pi i times the integral of z f prime of z over fc dz Well the problem is if the function is not quite periodic then the left and the right intervals no longer quite cancel out So what we get is something like omega 2 times the integral from 0 to omega 1 of f prime of z over fc dz Minus omega 1 times some integral and this is the logarithm of f of z and it might change by log of c1 And if c1 is an nth root of unity this might be some integer of the form m over n times 2 pi i So here we're taking c1 to the n equals 1 We're taking n equals 2 in our case And if we work this out we find that this is actually not in L So this is not in zero it's some element of c modulo L It's not necessarily the zero element It's in 1 over n L modulo L if c i to the n is equal to 1 for some integer n which is the most useful case So instead of the sum of the zeros minus the sum of the poles being zero modulo L It's some fixed non-zero element of this lattice depending on this quasi periodic case So let's look at the case of the Jacobi functions In this case c i is equal to plus or minus 1 So the sum of the poles minus the sum of the zeros is equal is something in half L And you see that there are four possibilities if here's a fundamental domain And there are four elements of c modulo L such that two of that is in L which are like this And one of these doesn't count that's just the periodic case where you can't have a function with one pole So we can have functions and so these are almost periodic functions with a pole at one of these three points So these four points So the sum of the pole minus the sum of the zeros is at one of these four points So we've got a way can we get a function with just one pole of order one And the answer is yes we can and where are you going to put this pole Well it's obvious where you're going to put this pole if you've got a function with a pole you put it at the origin You wish in fact Jacobi decided not to push it at the origin for complicated historical reasons He actually put the pole here at omega 2 over 2 and this is another way in which Jacobi's elliptic functions have kind of got messed up Jacobi put the pole in the wrong place So the pole of s n c n d n is at omega 2 over 2 and not 0 for some weird reason So what's going on here is the poles and the zeros look like this So here's a fundamental domain Omega 1, omega 2, 0 and the functions all have a pole this point here So that's s n c n d n all have a pole here And then the sum of the poles minus the sum of the zeros must be in a half else So there are three possibilities for 0 and s n is equal to 0 at this point here And c n is equal to 0 at this point here and d n is equal to 0 at this point here By the way I should say the periods are usually called 2k and 2k prime Instead of omega 1 and omega 2 when people do Jacobi functions As I mentioned earlier one of the annoying things is when there are several completely different systems on notation When people do Jacobi functions they use a different period lattice from people who do viastrice functions Which is a utter pain when you're trying to compare the two sorts of functions Well so we want to find functions with a pole of order 1 here and zeros of order 1 at one of these three points So let's find existence and relation to the viastrice function So at the moment we haven't yet actually shown that the Jacobi functions exist Well existence is quite easy if I take the viastrice function of z and subtract rho of omega 2 over 2 This has a zero of order 2 at omega 2 over 2 and a pole of order 2 at zero And it's got no other zeros and poles So because all its zeros and poles have even order we can take its square root and get a single valued function So this is now single valued I mean usually when you take the square root of something you get a horrible mess whenever the function vanishes because you get a branch point But if all zeros of order 2 then taking a square root just gives you a zero of order 1 And it has a pole of order 1 at zero and it's periodic up to sine So if we define this function here then we see that f of z plus omega 2 is equal to minus f of z And f of z plus omega 1 is equal to f of z So we pick up a minus sign because we're sort of taking a square root which puts in minus ones at various points This is almost but not quite one of the Jacobi functions You see the Jacobi function has a s n has a zero here and this one has a pole at the origin So in fact this function is sort of linear in 1 over s n And some constant plus a constant times s n I'm not going to put the constants in because I can never remember them And the other problem is people use different conventions and different notations So whichever constants I put in someone's going to tell me I'm wrong So that constructs the function 1 over s n So you can obviously just take an inverse and get the function s n and you can get c n and d n by similar formulas So that shows the existence of the Jacobi functions with the properties I've stated And then you remember that all elliptic functions for L are given by either by a stress p function or its derivative Let's take functions with the only poles at zero And all elliptic functions are given by quotients of these and there's a similar theme for Jacobi functions If you take all elliptic functions for 2L they're given by poles only at zero They're given by c of n s, c s and d s And I'd better explain this notation so n s is equal to 1 over s n c s is equal to c n over d n where these are Jacobi's functions and d s is equal to d d n over s n So these are the functions these are three functions with poles at The origin and there's also a relation between these functions here you remember there's this differential equation Satisfied by Vistris's function which is minus 4 p plus g So p cubed plus g2 p plus g3 Similarly, there are some relations between these three functions In fact, if you look at n s squared, c s squared and d s squared These all have a double pole at zero and no other poles so they're all linearly dependent on each other So we should really quotient this out by various linear combinations of these These are usually written in terms of the Jacobi functions s n c n and d n as follows so s n squared plus c n squared equals 1 And k squared s n squared plus d n squared is also equal to 1 By the way, you notice this looks rather like the formula for sine squared plus cosine squared equals 1 In fact, sine and cosine are kind of degenerate cases of s n and c n if you allow one of the periods w2 to tend to infinity This seems to be one of the reasons why Jacoby put the pole in such a funny place He was trying to make his functions look like sine and cosine And sine and cosine definitely don't have poles at the origin So if you want your functions look like sine and cosine, you'd better put the poles somewhere else We can also use this to give several embeddings of elliptic curves For instance, Weistreist's function we saw maps the curve c over l to a cubic in the projective plane Similarly, by mapping a point z to n s z c s c and d s c, we map the lattice c over l to a degree 4 curve in e cubed And this again shows that Weistreist is a little bit easier than Jacoby because a degree 3 curve in a 2 dimensional plane is a little bit easier to handle than a degree 4 curve in space I'll just finish with some historical remarks about why Jacoby chose such funny conventions So we can say why did Jacoby put the poles at omega 2 over 2? Well, as I said, he had three functions s, n, c, n and d, n And these are actually short for sine of the amplitude, cosine of the amplitude and derivative of the amplitude Where the amplitude was a certain function that Jacoby thought was really fundamental Well, you can see what the amplitude is because d n is its derivative So the amplitude of z is equal to the integral from something to z of d n of z d z The problem is this function is rather a mess, you see it's not quite periodic So amplitude of z plus 2 omega 1 is equal to the amplitude of z plus some mysterious constant of integration And similarly the amplitude of z plus omega 1 plus omega 2 is equal to the amplitude of z plus another constant So the first place it's not periodic, it's only periodic, adding a constant The other problem is it has lots of logarithmic singularities Because d n is poles of order 1 and when you integrate those you get logarithms So the amplitude is a multi-valued function of logarithmic singularities and branch points everywhere It's really quite a mess, I've got a picture of it here So here's a sort of picture of the amplitude value of the amplitude function These sort of black patches here are where the function has a branch point and becomes multi-valued if you go around these points here And as you can see the function really looks like a bit of a mess So my feeling is that the amplitude function is probably best forgotten about it It's mainly of interest for historical reasons that it's how Jacobi actually originally came across elliptic functions You can define it explicitly not using d n, in fact this is the way Jacobi originally did it You write z is equal to the integral from 0 to phi of d theta over square root of 1 minus k squared sine squared theta And then phi is equal to the amplitude of z So the amplitude was really a bit of a mess First of all you've got this rather weird looking integral And secondly it's not given by the value of this interval, it's given by the inverse of this interval The amplitude is what you have to integrate up to in order to get the number z So you're dealing with this sort of weird inverse of an integral I think this kind of illustrates the well-known saying that pioneering work is really clumsy The first person to do something quite often gets things in a rather clumsy way Because they're doing something the first time and don't know what should be going on And then later on other people come along and clean up the area So Weistreist came along and gave a much cleaner version of elliptic functions But we should still give the credit to the person who came first and did things in this rather clumsy way So Jacobi's elliptic functions like the Weistreist functions have absolutely masses of identities I'll just show you very quickly show you a few of them Again wish even Watson is absolutely pages and pages of these You see there are all these identities there are so many identities that they write s1 instead of sn of z And it just goes on for pages and pages I don't see any point in going through any of these identities You get things like addition formulas which are rather similar to the addition formula for Weistreist functions and so on But all of these identities or most of them can be proved in a sort of mechanical way just by checking where the poles of both sides of the identity are Because if two elliptic functions have the same poles they must be the same up to a constant Jacobi didn't seem to have this technique I mean one of Jacobi's problems is that he was developing elliptic functions before complex analysis had really been developed So he simply didn't have things like Leuville's theorem And instead of giving these very simple proofs of identities just by checking the poles He gave rather complicated proofs by sort of explicitly calculating both sides and showing they were the same So to summarize the difference between Weistreist and Jacobi is Weistreist was looking at sections of an order one line bundle By sections I mean Meromorphic sections and an order one line bundle is just a fancy way of saying a complex valued function Whereas Jacobi the Jacobi functions are sections of the three order two line bundles This is all over the elliptic curve C modulo L So Weistreist and Jacobi are the cases n equals one and n equals two of looking at line bundles of order n As I mentioned earlier you don't have to stop at n equals two if you really want to you can do sections of the nine or eight order three line bundles And while away many happy hours writing down masses of identities between them but I really don't see the point Okay next lecture will probably be about theta functions of lattices


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Hey everybody it's Sam at Mixed Up Craft, thank you for watching my tutorial today. I'm going to be showing you how to make this really sweet double flip lid box and this is basically evolved from the chocolate box that I shared for Mother's Day. I had a lot of requests to do a more of a box style so although that was a box it was very flat and it was I guess it was quite specific for that tray of chocolates. So I've decided to do this one here, it's a great size. This one measures four by five by four and you can see how it looks on the side there and then you just open up the two sides drop down and inside you have this lovely golden chocolate Easter egg. This is from Marks and Spencer's, you can buy them on their own and that is just nice then because you can make some packaging yourself. I've put some of the shredded paper here and you've got lots of room to add other little treats in there and of course you don't have to have this as an Easter theme. This would fit a nice candle, other nice gifts inside and you could decorate it accordingly but I think it's very very sweet. I love this little topper which I've made with the little carrots and I've put the little rabbit there as well. It's very straightforward to make so let me show you how. Okay so these are the papers that I downloaded and printed off so I printed some onto my 300gsm cardstock and the best way to do that I've just got a standard printer but if you put one page in at a time then you'll be able to print onto cardstock. Depending on what printer you have if you've got something that's a lot more advanced and you can probably put more in but I've had people before ask me how I managed to print onto cardstock with a standard printer and that's just how I do it. So just go into the image print just the one and just have that one piece of card in. Let it pull it through and then if you want more then just push print again but I've done the polka dot. I went a bit off there but it's fine I can still obviously use that one. That's really lovely and then these ones I've just done onto paper because I thought they'd be nice to fussy cut. So I've done two of those and then I've done this on cardstock because I think this is going to make another really nice box. I can always print more of that one which I probably will actually because it's directional. I'll probably use two pieces and then I've done these on cardstock. Again I'll probably end up fussy cutting these and I might fussy cut a couple for the topper on this project but I just think they're really really sweet. They're from the papercrafter website which I will link in my blog post and I just think you know I don't tend to really buy a lot of Easter papers and I've not really I've struggled to really get some for this series this year so I've just had a look online and there's so many just type in free Easter downloadable paper and you get a ton of websites come up so yeah that's what we're using and there's also this gingham. There was one other which was the same as this one but it was in like a stone colour that one there but I didn't really like it. It was a bit too I just thought it didn't go with this collection as well as all of these because they're so bright and colourful but I love the gingham ones that's what I'm using today so I've already done one half just like I did when I done the chocolate box and I'd started to cut into this one because I just wanted to check that everything was going to come together so as I mentioned at the beginning this is all using your letter paper size or your A4 card stock so you will need two pieces that are nine and a quarter by seven and a quarter and this is for the box so on both pieces you want to score along the nine and a quarter side at three and three quarters and eight and three quarters which you should have a half inch tab then rotate and then score at three and a quarter along that seven and a quarter side do that on two pieces and we're going to cut it so it looks like this and then you will need two pieces for the lid that are nine by six and a half and along the six and a half side you want to score at two and six so you'll have a half inch tab here then rotate and along the nine inch side you want to score at two and seven so you'll have two two inch sides okay do that on both pieces that's everything so I'm just going to get rid of my scoreboard okay so first of all we will concentrate on the box so you want to fold and burnish all of the score lines and then with your scissors you want to have the half inch tab on your right hand side and you're going to cut up the score line to the first score line like so and then again with this one here just cut up that first score line that score line to there and then just remove that piece completely so you have something like this then with the tab you just want to take a little wedge off of each corner and then the shorter side is the side of the box so this is going to come under and that's going to become the front or the back so it's this smaller section so the left hand side that you just want to take a wedge off of each of those corners just so you don't get anything overhanging okay that's a bit messy but you're not going to see any of that it's because I'm using these scissors I should be using my nice longer ones here which I can just then snip away in one cut much much better there we go okay so you'll have two like this next well I didn't take a little bit off of the top of that one what you want to do is add some glue onto one of the tabs and then you're going to stick this piece over the top just on the score line this base score line here make sure that stays nice and straight and all lined up and as I always say if you've got anything overhanging at the top at least then you can trim it away okay then flip it over fold over this half with the other tab and add some glue onto there make sure you don't get any glue on the box itself and then fold this one over and it will all line up again just make sure that those score lines all meet and just burnish everything okay so you'll have something like this next turn it upside down decide which you which you want to be your front and back and then just fold for me I'm going to fold the back down here I'm going to add glue all over here okay let's just fold those both in and then add glue all onto this bit don't worry that you've got the gap there you can cover it with some card stock if you wanted to but you're not going to notice it's the base it's just the it's the size that I've done and I wanted to try and maximize the box size so then just hold that down in fact you don't need to put the glue on that bit because you're not sticking anything to it but that's the nice thing about this callow you see there when you rub it it just all comes off comes off into a bit of a clump so you can just but it doesn't ruin the card stock okay we can just go in there and just make sure that's all sealed down so I'm probably going to line the inside anyway because I just like that and I think for this kind of project because the lid opens up it'd be nice so I'm going to decorate the bottom but I mean if you want to decorate the bottom there as well you can do so that's the box done then we're going to make two of these now I had to add my hinge because I cut it off by accident but basically that's going to stick under there just like we did for the chocolate box and you can see how one side's going to go over like that it looks really nice so to get that you've done all your scoring so again fold and burnish the score lines okay so you want to have your half inch tab on the right hand side ignore that I've cut all this but have it on the right hand side you'll have your two inch score line here and this is that six inch score line there you want to cut up this one just to the first score line so I'm just cutting up just to that one there sorry I know we've got the gingham it's a bit busy but remember it's that two inch score line you're just cutting up to this two inch one because along here was the two and the six okay and then you'll have this little corner here you just want to cut up and remove that piece completely because that's going to be the hinge that we stick underneath the box okay and then flip it around so now the tabs on the left hand side and again you want to cut up that two inch one to that first score line and then this one here and because I've already done the other cut that will come out straight away you'll have something like this I'm then going to take a wedge yeah that is right off of this piece and also off of this one here what you will have okay and you'll want two pieces like that next you're going to add a glue onto this tab and you're going to stick this one under here and bring this one around and they should join up nicely again just spend a minute sticking that all down add glue to the top of this one and stick it down the same way then you want to cut from the bottom here across to here now if you want to draw a pencil line from the top left down to the bottom right you can but I've got my long scissors here and I'm just going to go in and just cut this freehand just eyeball it there like so okay so again I can come around this side we'll just make sure that's all stuck down like so and then you want to add glue under onto your tab again you might want to just take a little bit off of the corners of each one there actually because you don't want that sticking out and then take that one off it'll fit along the long side if you pop it on there and then turn it upside down you can really push down on that tab and make sure it's nice and flush against the side there and again if you want to decorate that once you stuck this down because then you can cover everything actually I might do that we'll see at the end in fact this tab will end up covering that section anyway so and again yours won't have this additional tab yours will all be like that first one I'm just going to stick that over it will overlap the other one slightly and bring that over when that's all secure flip it over and you've got a flip lid box or double flip lid box you close it up whatever one you put down first the other one will just sit slightly over the top it's a really nice size box so now we can start decorating everything okay so I've just been having a play around with some stuff I've cut some circles I'm going to use but inside you'll see I've put my golden egg and I've just used some of this straw or well just shredded paper to act as straw and it's this one here which I'm sure I did share in my what did I get because I shared a lot of the Easter stuff that I had purchased yeah shredded craft paper so that's all in there and I've also lined the bottom you can see there just as some polka dot paper so if someone does want to use it afterwards then they can and I've done that on the bottom as well so you'll want two pieces if you want to decorate oh that's slipped oh dear see it's wonky because um obviously I didn't uh I moved you know I carried on whilst it was still wet and it's moved so never mind it's three and five eighths of an inch by four and seven eighths of an inch so it's a bit wonky on the bottom there but it's the bottom so I'm not too worried but I think that looks really nice and there's room for lots of other things as well I'm keeping mine white on the sides but you could easily add some you know pattern paper here as well but I'm going to have this one on top it's going to stick to one half so whatever you know the the top side so this one's going to be on the actually no because I like it to open from the left it'll be like that so it's going to stick on the left hand side and then I thought it looks really cute with the three carrots on top I just love that but then I thought do I want to have the rabbit as well so I don't know whether to kind of maybe have the bunny behind a little bit um or maybe just kind of stuck resting on the side there like that I think it's quite a cute top and I can always add a sentiment I was going to bring in the little chick but I thought carrots go better with the rabbit so we're we'll scrap the chicks they're going to be used for another project so I'm going to just I've got my hot glue gun on these were all picked up from the pound store they do them every single year so I've got a few from last year and then I picked up some well a few bags this year because I've got these ones are felt but there's ones that I've got glitter on there's I've got one that's slightly bigger but they are all going to be in my shadow box which I'm currently making and then that one there so let's have those three kind of like that and then this one can kind of stick on the ends there quite a fun little topper and then that's going to sit on one half so I'm going to use my Kalao just a cover because this will really strengthen the lid make sure it's nicely make sure it's nice and even the circle sizes I've got three and three quarters for the larger one actually I'll show you them here well that's just kind of dry drying the circle sizes are yeah three and three quarters and then the next one down is about three and a quarter okay I can't remember where I got these from I think these ones were from the works if I remember rightly but it was a it was a good you know a few years ago now so but any nice circle you know you might want to do squares you might do something completely different but just make sure it doesn't shift like the bottom did and if I open that out I can just really make sure that sticks down you just got so much room in there so that's stuck down that can all come up and then that closes up on top isn't it sweet I just really like this I just wanted something that was just with the gingham and the orange and just quite clean and crisp I always like to go over the top with things so that's what I've done with the topper but I'm keeping the box all you know quite plain and then when you open it up you've got that golden egg inside I'm tempted to do something in here though let me just see what scraps of the yellow I've still got so I've just stuck these down here so it was the paper that was going to fussy cut but it's got a happy Easter on it so this is four and three quarters by three and three quarters check that one yeah that's four and that will you know just strengthen it some more and then it all closes up so you've got all the detail on top of there open it up and you've got the golden egg which I don't know if I said is from Marks and Spencer's so I love those ones that you can then make your own packaging for but they have it I think it's lovely so I hope that's you know helped those of you that asked for the double flip in a more of a bigger box I think that's a great size I think you can you know put a lot of things in that definitely a nice candle and because it's it's quite reinforced with the amount of layers that go up around it so you can you know put something a bit more weighted in there as well so as always thank you for watching and please give the video a thumbs up if you have enjoyed today and consider subscribing so you get to see more thanks for watching bye


TLO was pop We are on twitch. We are not live, but you can leave a like comment subscribe turning a post notification bells Let's continue to go to finally from Chicago to the UK And don't forget man if you like this videos man here like man if you like these reactions The best thing you could do hit like matter of fact it's shaped like this look If you go straight across from my fingers, so Anyway, and you take funniest moments ever This is more up-to-date this one the last one I watched yesterday Had old footage, but this is the new one.  He's definitely cabin.     You can both be think of produced cheaply in China It's dangerous for you. Can you believe that as you talk about vapes?  It might actually be bad for you That's incredible But that that surprised me like I couldn't believe I don't smoke anything.  What's that thing in school you learned with a how they refine oil?   What is he doing out there, man?  I have a lot of muscle It's it's painful to carry around.   I'm gonna be like bang.  She pulled out of a junction turned to turn right instead of just turning right She did some kind of you turn back into traffic and crashed It didn't make any human sense. I can't I can't explain it to you in a way that makes sense I know you're sitting there thinking no that can't be right. No, that's literally what she did traffic's going this way the traffic stopped Traffic she comes out said it's just turning right and going on her way She comes and just crashes into this car and I sat there and I thought How are women allowed to drive?    It's four and a half.  Yeah, like y'all wild and That's beyond me. Had he just what was the question? It's four and a half.  Yeah, I love it. I'm sorry. I'm sorry.  Oh, how are you gonna be PewDiePie?    Let me share my stomach. Hold on. Am I fat?     What's wrong with you? These are facts Go hospital later. Have a drink cigarette a cup of coffee Back in the game he's by this shit And now the other people don't listen to The exaggerated stuff he'd be saying he'd be saying some factual things though But did some stuff he'd be saying and he knows it it's it's going for a show It's exaggerated for tweeting on the news.   You don't need a vape Have you ever seen anyone with a vape? Have you ever sat in a room? So we go wait.   I'm risking cancer to look like a mafia boss fine I'm not sitting there smoking cigars and going I can't believe this is bad for me For me So she's already that's what I would have been like, no, what do you mean for you? It's not all about you. We're a couple and we have a public image which you have destroyed This isn't all about you that I would have interrupted her right there.  Hi, my name is k jankins. I am 2016 this the this the good stuff. Here we go. Look. It's elena. Pa. I'm years old I am a luxury real estate advisor But I also just started an only fan. So if you want to see me naked subscribe There you go Um, and then I uh, I have multiple boyfriends, but I'm still single. So life is good Okay, listen to me.  You're upset.   Don't walk around with a sword. That's what they want you to do In the uk literally goes against that people pull swords out their pants all the time. I'm gone


Hey, everybody, I'm CeCe. And the most interesting thing that I learned this year was that if you give me a good enough story that has enough interesting parts, I can be motivated to do things that I used to absolutely loathe. For example, running. So ever since I was a little girl, I just hated running for two reasons. First, sweating is gross. And second, I get really bored, right?  Doesn't work out very well. Now, as I got older, I tried again and again, because I've heard about the runner's high. It's so great, and the exercise is great. Then eventually, I came with the conclusion that I need to assure myself this will never work. But about a month ago, I started playing a game. It's very aptly named Zombies Run, which is kind of cool, right? The game itself is just an iPhone game, and I don't think that this slide is going to advance any more, but essentially what happens is, now I run about three to four times a week and two miles a day, right? So that's kind of crazy, and I don't know that I quite grasp that this is happening, but the way it works is through sort of a motivational story. All they really do is they take together audio files, and they mix it in with your own music, and suddenly when you push play, you become the main character in a zombie apocalypse story, okay? Now, to give you guys sort of the full experience, I've decided that I'm gonna re-enact the beginning of episode two for you. Now, I first have to catch you up with what happened in episode one. So imagine you're on a helicopter, and you're flying to this town to give them supplies, when all of a sudden, somebody shoots a rocket launcher at you, and your helicopter goes down, right? So you parachute out, you get on the ground, you know there are zombies around here, and you book it. Now, someone is talking to you on your headset. You don't know who, he claims he's from the town, but he doesn't know if you're alive or dead, so he's just talking, and that's when the episode starts. So this is really where I invite you to sort of close your eyes if you wish, and listen, pretend like you're running, and my voice is in your headphones. Wow, there's someone alive down there. Hey, hey, can you hear me? No answer, still look at him go, heading for that tower, just like I said. Okay, running person, if you can hear me, you're doing really great. The main pack is behind you, and you're gonna come out of this forest really soon, but there's a, well, I can't think of a better phrase than small army of zombies coming to get you. I don't do very well under massive pressure, but that's okay, you're gonna be okay. If you just head towards that large building to your left, everything will be okay. Hey, look at that, they're changing directions. Oh my God, you can hear me, okay, okay, great, it's cool, we're gonna get you safe, we're gonna bring you in, and no, we are not gonna ask them that question because they might be injured. All of the more reason to ask them that question. Listen, my name is Dr. Abel, and I'm the medic here, the only medic at Abel Township. Your path is going to take you almost to the old hospital, and we know there are medic kits there from the first wave of infection. If you could pick up just one or two, that would help us, but it's too dangerous, you know what happened to runner five. All of the Zoms are headed towards the site of the crash, but what about whoever fired that rocket launcher? Listen, if someone is gonna kill that runner, taking an unusual route will make it harder, not easier. I don't wanna be hard headed, but everyone in this township earns their keep. You should be able to see the hospital by now. If you get there and you can't find anything, we might not be able to let you in when you get here. So that's the beginning of episode two. It's pretty great. The game sort of takes you through some twists and turns that are like, not plot devices, but they're very interesting because the danger is never the same. It's not always zombies running after you. What's happening sometimes is let's say, hey, I have a running partner now. She doesn't believe me. She doesn't believe that I got off that helicopter and that I sort of killed somebody, took their clothes, and then came to this town. That's dangerous.  So what's really cool and what really sticks with me about this game is how much my survival instinct really just kicks in, even though all of my other sensory perceptions are telling me otherwise. So let me just paint this picture for you. Sometimes I like to run at night in a gym on a treadmill. And I get there, I push play, I'm on a treadmill. I know for a fact there are no zombies in this gym. Now, I see the TV screen in front of me. It is always a cooking show that I always cannot figure out how to turn off. It's night, so the window sort of reflect me back to me so I see myself running on a treadmill, right? I have my gym shorts, my headphones, my iPhone in my hand because that's how I do it. And all of a sudden you get to that point in the story in which there's some groaning and some moaning and it gets louder and the danger is totally there. And even though my very logical mind is telling me there is no way there's any zombies in this gym, I feel that urge to run faster because I have to and I do, and I do it over and over again. Thank you. Thank you. Thank you.


So welcome back folks, you may remember it. Now we get to talk about the importance of dead things in the forest and I'm going to start by looking at this wonderful example of a large snag that's a dead-standing tree and if you look carefully up here there are all kinds of holes in the tree. Those are holes that woodpeckers are making in the tree to try to find insects. They might also be holes where they're trying to build a nest. They often they build their nest right in the cavity of this dead-standing tree and so the thing to remember about snags is that the bigger they are the better they are and my friend Andy is going to help measure this particular snag and tell us that it's almost, what did you say? Let's call it 16. 16 inch diameter all right and so that is big enough for something like a pilliated woodpecker our largest woodpeckers to actually use. Now in addition to these large snags we have smaller snags in the forest that can be used by smaller woodpeckers like downy woodpeckers and then once those downy woodpeckers build a hole chickadees or tipmice will come in and use that cavity for their nest in future years. Something like this size would be good for a downy woodpecker or chickadees but definitely not pilliated woodpeckers. Now the next thing we're going to look at are dead wood on the forest floor. All right so if you look around on the forest floor here you'll see all kinds of dead wood on the ground. Now some people look at that and they think oh that's messy I want to clean it up make it look almost like a part but no no no we don't want you to do that we actually love down wood on the forest floor. We say messy is good and more is better. So here's an example of some down large down woody material this old tree that fell and is lying on the forest floor and in order to count it as being part of the down woody material we need it to be at least six inches in diameter and Andy what have we got here? In fact we've got more than eight. More than eight inches so so this is a pretty good size piece and it's also over 17 feet long. So this is the kind of log that a roughed grouse might stand on top of and flap its wings and send out a booming boom boom boom boom boom boom sound to attract a mate. It's also the kind of log that a fissure cat might run along the top of or look underneath searching for small mice and amphibians to feed on. So these features on the forest floor are really important. The other thing we want to look for are what we call fine woody material smaller smaller logs that are not logs smaller twigs that are on the forest floor but often in piles so not really what you see right in front of us but those piles of smaller twigs are places where birds can hide from predators and they can also look for insects that collect in that pile of force. In fact just last week I was out walking through the woods behind my house and there was a winter wren that was tucked right in the center of one of those piles of fine woody material. So we'll see if we can find one of those piles for you in a minute. So here's an example of a small pile of fine woody material and if you were to add a little bit more some other branched tops of trees and branches on top of this you'd have a nice little cozy cave where birds especially sparrows and wrens can come in thrushes can come in and hide in there from predators and they can also feed on the insects that collect in among these branches. And then another important part of the forest is what's here on the forest floor.  Now that you've had a chance to learn all about what different habitat features different birds need to rest feed and raise their young. We'd like to invite you to come back and talk with several professional foresters about how you can put together a management plan that enhances habitat for birds and other wildlife along with meeting whatever other goals you have for your property. So please come back soon.


Hey guys, it's Jen for my create crafts in today's video I'll be showing you step-by-step how to create these Halloween rolling pins They are great for any decor and would make a great gift to give someone or even sell at a craft fair or in a craft store They costed me a little under a dollar each to create and they were so much fun to create now. Let's get started After so to create these Halloween and autumn rolling pins you want to start in design space And I'm gonna start in the left-hand side here where it says text and I'm just gonna type in a whole bunch of things that I Want to cut out so I have on my mind. I have spooky Halloween trick-or-treat candy corn hay rides and pumpkins You can add to or subtract any of those that you want, but I'm just gonna start with spooky oops and Then I'm going to do the same thing. Whoops. I was wondering why this looks so funny There we go guys. I know how to spell I'm gonna do the same thing. I'm gonna go to the text here and then just type in Halloween And I'll move it over so I show you what it looks like go to the text again try type in trick or treat And I saw this somewhere where I'm gonna bring this out where it says trick and then obviously or treat But I also saw it where it had a little line in between each one of these and I thought that looked kind of neat So I'm gonna try that and I kind of like how that looks or maybe just get rid of the extra Spacing here and see whoops see how that looks. I mean, it's all on how you want to do it I just saw it and I thought it looked kind of neat that way So I'm gonna leave that one to so I'm gonna go to text and I'm just gonna do candy corn and Then I'll just do maybe one more and show you what it looks right looks like I can talk to you guys my goodness Hey rides. There we go. And just for the heck of it. I'm gonna do another one says pumpkin Just because that was on my list So the next thing you want to do is maybe not you may or may not want to change the Fanta on this I actually do I do not like this one. I have one in mind So I'm gonna go up to the font box up here and I'm gonna type in skinny And here it is the skinny It's called and this is just a regular one that you get from a design space I'm not sure if it's one that you have to pay for if you don't have the yearly or the monthly subscription I actually do pay for the yearly subscription. So that's where I'm getting this from You could also go to defont.com and pick out a font that you like You just have to make sure that if you're going to be selling these rolling pins that you buy the commercial License for it. Otherwise, you could get in trouble. So I have a video on that how to download defont Fonts from defont.com and put it into design space if you're interested in that So I'm actually gonna leave the spooky one. I'm gonna find a different one for that one So I'm gonna do all of these. I think I really like this This skinny font, so I'm gonna go back up here and just click on it So all I'm doing is clicking on it going up to the font and it's already saved it from the last one that you've already done So I'm gonna Okay, that doesn't look so good anymore guys, so I definitely do not like that with that in there So I'm gonna take this out And leave it that way. I don't know. I guess it must have been the font that it was so for the spooky one I'm gonna do the same thing go up to the fonts get rid of that one and just kind of look quickly for you know Something that jumps out at you. I'm just looking for something. That's bold. That's a little bit too bold and Kind of scary looking I guess we're gonna see what this one looks like Something that's more bold is what I'm really going for so. Oh, I do actually like that one So this was the BFC blueberry pancake while it sounds good right now. So I kind of like the way that one looks Yeah, I like that one and then I'm just gonna do the same for this Halloween one I'm just gonna randomly find one you might have one that you have in mind But I'm just randomly going through this really quick to see Oops, and I like this one so far It's called dreamland, but I saw that if you use it you have to pay the 499 I think it was here right here 499. So no, thank you I will keep scrolling to see if I can find something that is free again You can go to defont.com and find something there But I actually like this one. So I think I'm actually gonna leave them all that the way that they look I love this spooky one So for these I figured for my rolling pins, they're seven inches long So I'm going to actually do a four by point seven five and see how that looks So to do that you want to select this Unlock your lockbox up here and then change your width to whatever you want I'm gonna try four by point seven five and see what that looks like not too bad So I'm gonna do that same thing with this unlock it and then change it to be whatever you want Mine's gonna be four by point four four by point seven five Guys I'm having troubles talking and the reason I'm doing point seven five instead of one inch long is I wanted to be High as I want to be able to see it. So I don't want to go too far So I'm gonna do this one really quick show you four by point seven five And I'm just gonna go ahead and fast forward this you don't have to watch this But I'm just gonna do the same thing Okay, so I changed all of these to the four by point seven five as you can see up here so the next thing that I'm actually gonna do is Change them all to be black. I think I'm thinking about it. Maybe I'll do a few white and a few black because my colors are going to be I'm thinking black maybe red some sort of green and maybe an orange So I'm kind of thinking what would go with that So maybe candy corn we could do white and then we could put it on like an orange one Hey rides is kind of like, you know Kind of more of a white color right instead of doing like a let's say tan or something We'll do that and white as well trick-or-treat.  So it all depends on what colors you have on hand you actually could go with this and Change it to be a trick be orange or the maybe like a Purple and treat could be black But I'm just gonna actually leave this the way it is trick-or-treat spooky obviously has to stay black, right? So I'm gonna change the pumpkins to orange and you could do whatever color you want You do not have to pick any of these colors that I am doing That's it So I'm just gonna go up to make it up here and I'm gonna show you which setting I always like to use when I'm using just regular a 651 vinyl So it's gonna show it to you here So I'm just gonna move it over just a little bit that I can get my scissors in between each one of these Just move it over just a tad But I love how this looks ready. Then I'm gonna go to the continue button and I have the maker machine. I absolutely love it. If you guys are upgrading I would definitely look into the maker I know that they have a maker 3 but I think really the only difference is as you can put Longer material in it and it's a little bit faster. I think that's my understanding of it So I am in no interest of upgrading mine But if you have like, you know, like a explore or an expression and you want to upgrade I would definitely look into getting the maker. So here are my base materials I always used a stencil vinyl one here. I don't use a lot of Cricut brand I just use whatever is on sale and whatever I can find So I will leave all of my supplies in the description below pretty much everything that I buy I buy from Amazon I don't head out too often to buy things I kind of just get the idea and go ahead and purchase it on Amazon really quick So I will leave all the links below to the things that I purchase.  I will cut this out I will weed it out and then I will show you the next step of Painting your rolling pins and then how to apply this to it. Stay tuned guys I can't wait to show you the finished product. So while my SVG files are cutting out I'm gonna show you how to paint these. So I just bought these from Amazon. They're just little mini rolling pins and I'm using some chalk paint which which I absolutely love. I love using this It's just a Waverly chalk paint that will come in focus. There we go You can choose whatever colors you want But I'm just gonna go with the you know traditional fall colors Just this green a little bit of a red and then an orange as well And then I also have black too, but this is not a Waverly paint So since I have a crafter my kids think whatever's mine they can use So unfortunately, I do not have any but I did find some of this black stuff The reason I really love this Waverly chalk paint is it's a lot thicker than regular paint And it dries extremely extremely quick So when you put a coat on here by the time you're done with this one and you go to the next one Your coat will be dry on here.  I thought they were so cute So the basically all you want to do is just take any kind of paintbrush. I just bought these paintbrushes I think I got these at Hobby Lobby and like 40% off or I have a huge bucket I will show you guys really quick of Other paintbrushes and such in here. I have these little foam ones. I have down here Some little makeup brushes little makeup sponges as well So I'm pretty quite prepared with this so I'm gonna show you this really quick So you just choose whatever color you want and then just basically you can just paint on the reason I'm using a smaller paintbrush here is because I want to get the edges on here I don't know if you can see it really well But does have like a curve to it and I kind of want to end that curve So I'm just gonna choose we'll start with green here first and just kind of go on here and You can use like painters tape on here if you want and just kind of get a straight line Or you can like paint the whole thing if you want, but I'm just taking it and kind of go into the edge as far as I can So there so I got just the edge part right there and it's up to you if you want to do the whole thing I mean you can definitely go ahead and do the whole thing But I kind of just want to do the edge part So I'm just gonna take it and then just go all the way down chalk paint dries extremely quick It's already dry. So I mean you can put it down if you want it. So there is that one So I'm just gonna leave this paintbrush here Let's see. I'm gonna put on top of a cup over here on the edge So that's not flat on anything If you're gonna stay for me There we go and then just do the same thing with the next one So like I said if you want to do the whole handle you're more than welcome to it Whatever you want to do in your project, you know You can do a different color on the handle and a different color on the actual roller itself So I'm just gonna fast forward this part and finish this one up And then I will show you the next step of applying your vinyl stickers to it I will have them weed in and already finished kind of skip that process And show you the next step of putting these on and then how to finish them up and make them look really nice All right. So this one is finished. I love the color of this one. Here we go And I did the edges really well and I think you're in if you do make a mistake on the edges here You can most definitely put ribbon or something or raffia or whatever on the edges So you'll never even see it just as long as you're not covering the words that you're going to put in the middle So I'm going to show you easily how to put on your vinyl stickers on these so stay tuned guys All right guys. So my rolling pins are completely dry I actually did end up doing two coats on these I didn't realize how much the paint would soak into it So depending on how many coats you want to do it and how dark you want to do it Is what you're going to want to do. So here are the three colors. I chose once this one comes in. There we go And my daughter came in and did this lighter color green one, which I really like So I love it when they come in and help me do things. So here are the Uh Files that I did so I have this spooky one and I have this pumpkin one But you know, I couldn't find orange anywhere. So I just used a holographic kind of orangeish color It's kind of orange. It's kind of green kind of neat And then I did this candy corn one a halloween one trick or treat and then the hayride one Which you probably can't see too well, but it turned out really cute. So the next thing you want to do To finish these guys up is just take some transfer tape. I love this transfer tape. I talk about it all the time So there's a lot of dog hair on it. Um, I talk about this all the time. I bought this huge roll I think it was like $15 or something, but it goes a really really long way and I love it. I think it's the Fresca if I say it right and I can't remember what it's called I'll leave it in the description below in case you're interested in it So I'm just gonna take a little piece off of here and just gonna kind of eyeball it the longest one that I need And just cut a piece off. I know people sometimes use like the dollar store Transfer tape. I was using that for the longest time But then I was having problems with it So I kind of just went to this one and I love this one because you can use it over and over and over And it will not lose its stickiness. So we're just gonna start with this first one. It's a hay rides So I'm just gonna put it on here And then just use my little scraper and scrape it on to the transfer tape really well And then what I like to do is actually flip it upside down and then make sure all the letters come off onto my transfer tape Now is the fun part where you get to pick which one you want to put it on So I'm just gonna start with this orange one really quick ups and show you guys so Kind of fun how you want to do it here If you want to put like right in the middle if you want to put it on the side more I'm actually going to Take it and put it more towards the side The s is more on the side Because I think I'm going to put some like raffia or ribbon or something on it But then you want to make sure that you put it this way in the middle I know it's kind of hard to say in the middle But you want to make sure that your word is straight and you don't have it like crooked in any side So then you just want to use your fingers And then just push on here really hard And make sure that the vinyl sticks to your rolling pin I'm going to apologize for the noise in the background My door is open because it is so beautiful outside and I have a wind chime out there So if you guys hear that that's what it is So then I'm just going to peel back my transfer vinyl and if it does happen If the vinyl does happen to come up just give it a little bit of help and hold it down Um, but I'm using 651 vinyl. So this is a permanent vinyl. So it does take, you know, 24 hours to fully Set on here. Um, so I'm just using my thumb and as I'm pulling off the transfer tape. I'm just holding down the letters And once it sits for a little bit, like I said 24 hours, it will become more sticky Instead of work guys more sticky. It will become stickier. There we go And then I'm just going to save this transfer tape for the next one And then if it is coming up, you can just use your thumbs and kind of push it down Oh, wow, that is really cute Um, but I sometimes if I find out I'm having vinyl not stick to wood very well, you can use Um mod podge and put it on there and then it should stick on a little bit more But you have to watch out for the modge podge if you use the glossy kind It's going to make all of your wood glossy, which I don't think I would like it in this instant But here he goes Once it will start There we go Like I said, it's my husband's camera. So I'm rather new to this so and the full setup So I'm hoping this video is turning out really well. Is this video working well? Please let me know guys. Um, this is a new camera to me Um, but this is the first one I guess I'm gonna take like some raffa or something kind of wrap around the edge or even take some, um Really really thin uh ribbon and put it on the side, but I just want to show you this part first So there's the first one and then I'm gonna show you a thicker one. So we're gonna try this, um Spooky one. I really like this one this spooky one here. So I'm actually going to take I don't know should I do a red? Sorry about that now my dog's barking in the background or this orange I don't know which one will look better in here. It's kind of like your own opinion on what you guys want. I actually like I don't know. I kind of like the Orange a little bit more. So I'm just going to take the same piece of transfer tape and do the same thing I just did take my squeegee And rub it on here really well. Like I said, I love this transfer tape I will just use this over and over and just kind of put it to the side and Keep using it until it is sticky no more So there there's that part and then just do the same thing Just grab it and put it on here when I do it kind of backwards. So maybe you guys can see it better on camera here And you know now Knowing what I know now, I think I might have went a little bit smaller than the four inches on here Just because I would like to have put a ribbon right here on the side I also was doing some more research While this was cutting out And I saw that people were painting the handles a different color and I kind of like that So maybe I'll go back and you know paint this handle like an orange color Or go back and paint this one. Maybe the handles. I'll do purple. I don't know I think that was kind of neat. That would be like an extra step you could put in there Um, but I just think these things overall are so dang cute. So I'm just doing the same thing I'm just using my finger my thumbs and kind of pushing it on here rubbing on really hard and hopefully it will stick Oh, wow, this one's sticking a lot better. Maybe it was just the vinyl was too thin But there you just peel it off Perfect just like that Look at and that cute spooky my mom would like this She actually has a black cat and his birthday is uh, october 31st and his name is spooky So maybe I'll do that. I'll go back in here and maybe paint this black since it's already black here Um, but yeah, I'll show you what it looks like in the end, but I really like this spooky Anyways guys, are you liking this video tutorial? I hope so please give me a thumbs up if you are and hit that like button I love hearing what you guys have to say about my tutorials. Please let me know Do you like this one? What can I do different? Have you made these before? So I'm actually gonna go ahead and finish these up and figure out which ones I want to put on where So any fun ways you can guys can do this. Please give me a tip. What else would you guys do? I think I might go back and color a few of the handles in as well Um, but I really like how these look. Um, I'm gonna have to finish them up I'll show you what they got what they look like, but in the meantime Give me a thumbs up if you like this video Happy crafting everyone. Stay tuned to see what it looks like


I want to talk to you about margin. Someone emailed me about a trade that was in an email on Apple, which cost more than $100 a share. And they said, oh, this is very expensive. Not really. Why? Because most day traders trade on margin, depending on where you trade, at what broker you trade with, you could get a four to one margin. You could get a 10 to one margin at a proprietary day trading firm. So for example, I'm just going to use a very basic example. Say you have a stock that costs $100 a share, okay? So you want to take 500 shares of that. How much buying power do you need in order to do that? $50,000 buying power, not cash, okay? When you day trade using margin, that means you take the amount of margin you have. So it would be either four divided by 50, or which would be $12,500 cash. Or if you're in a 10 to one margin place, it would be 10 divided by 50,000, which would be $5,000. So you could buy 500 shares of a stock with a $5,000 cash account at a proprietary brokerage firm if you have 10 to one margin. Does that make sense? So you do not need all this cash in order to day trade the full amount of cash value all day trading. Or most day traders trade with margin, all right? So that's one, two. If you don't want to trade with margin, you can buy calls, which is an option. If you're going long a stock, which we went long Apple the other day, or you can buy puts if you're shorting the stock. I have an option in these letters as well. When you trade options, you just pay the cost of the position, which could be $1 or $2 or $3 on a stock like Apple, for example. And again, you can buy calls on a stock like Apple, for example. And again, you don't need anything more than the cost, whatever that cost is for that position, to buy the option. The difference between options is you can hold options overnight, but the options expire in a certain time frame, okay? And you also have to know how to trade options if you're going to do options. It's different from day trading. It's two separate ways of trading the same stock, which you can do. One is not on margin. One is on margin. It depends what you want to do. It depends if you want to day trader. It depends if you want to do options trades. With options and the benefit is you can capture overnight moves. With day trades, you can get quick moves and you can take larger sizes for fast moves that go in several minutes and be done very quickly in the day. So I just want to point that out because someone emailed me and I thought, gosh, you know, I have to do a video on margin. I can't believe that people actually think that when you take 500 shares of a stock like Apple or really anything that we trade that costs more than 100 dollars a share, that people think that people are trading these and have hundreds and hundreds and hundreds of thousands of dollars in their account or taking cash business and that's not the case. I'm surprised that people don't understand margin. I will talk about it more. I will put it more in the emails. But if you do not understand what I'm saying, one, you can Google and read about margin. Two, you can ask your broker. Or three, call me or email me and I'll explain it to you better if you don't understand what I've said here in this video, okay? The reason that people can trade, regular people can trade stocks that cost, you know, 100 dollars a share or 200 dollars a share, something like even the SPY, like the, you know, which is the ETF for the S&P is because they're trading on margin and that's how, okay? With regular accounts. It depends where you trade. You have to talk to the broker about how much margin you can get. Email me at Melissa, thestockswitch.com if you have more questions. Thanks, everyone.