Hello everyone, welcome to the talk. My name is Rishi Rajivathacharya and I'm going to talk about memory type reductions for some key encapsulation mechanisms. I'm going to start with the notion of a reduction. A reduction talks about two families P and Q. You can take for example, P to be the family of cyclic groups, whereas Q can be thought as a public encryption schemes. Each family has its own security game, say the DDH game or the INCP game in our examples, where they are hard to win against. The reduction comes paired with a construction C. C takes an element F from P and produces CF from Q. The reduction R ensures that if there exists any attacker A winning against CF in GQ, then the reduction with the help of the algorithm A can beat the function F in the game GP. As the algorithms are probabilistic, we measure the efficiency of a reduction in terms of success probability. We say our reduction is tight if for the same success probability as A, the reduction needs to spend approximately same time. Interestingly, the traditional definition of tightness that we have discussed does not take memory into consideration. However, memory consumption of the reduction is also a very important parameter. Consider primitive P against which the best known attack works in 2 to the power lambda time, where lambda is the security parameter when only constant space is given. However, if more memory is available, say 2 to the power lambda by 8, then the time requirement of the adversary suppose drops to 2 to the power lambda by 8 as well. Now consider the function F from P and the construction CF. Suppose the adversary A breaks CF in the game GQ in taking constant space and 2 to the power lambda by 4 time. A tight reduction would imply that the reduction R would win the game GP against F in time 2 to the power lambda by 4 as well. As we can see, if the reduction R takes more than 2 to the power lambda by 8 space, then the reduction is meaningless. On the other hand, if the reduction takes same time as A, then the reduction breaks the security conjecture of P, that is, it breaks the record of the best known algorithm for P. The issue is not imaginary. Many hard problems are indeed very memory sensitive. To remedy the situation, Orbach-Cache fashion case in crypto 2017 introduced the notion of memory tightness. In this case, we have a memory parameter in the reduction as well. We say a reduction is memory tight if in addition to the same success probability and the same time complexity criteria, the attacker and the reduction must have the approximately same memory as well. The natural question that arises is whether a memory tight reduction exists. In the same paper, the author showed that a memory tight reduction for existential unforgibility of RSA full domain hash exists. That technique formalized the notion of simulating the random oracle by pseudo random function where the key of the pseudo random function is sampled and the saved by the reduction. In addition, the reduction rewinds the adversary once. Apart from this, all the known results about the memory tight reductions are either lower bound or impossibility results. The starting point of this work is the open problem raised by ACFK, namely finding memory tight reduction for the in-CC security of hash L gamma KEDM. Let us informally define the notion of KEDM. There are three algorithms, key generation, encapsulation and decapsulation. The key generation takes the security parameter and produces the public key secret key pair. The encapsulation algorithm, which runs on the public key, produces the ciphertext and the key pair C comma K. The decapsulation algorithm takes a candidate ciphertext and produces a candidate key. The correctness requirement says that if C, which is input to the decapsulation algorithm is indeed produced by the encapsulation algorithm on a public key PK, then the decapsulation algorithm running on the corresponding secret key SK must produce the matching key. The security requirement says that given the ciphertext C, the key K is indistinguishable from a random limit from the key space. In this paper, we find memory efficient reduction for the in-CC security of hash L gamma. Our reduction works for two variants, namely the Kramer-Schub variant and the ECIES variant. For the Kramer-Schub variant, our reduction works for over all the gap groups. However, for ECIES variant, we require the group to have pairings. With an extent, this result to the Fujisaki Okamoto transformations, specifically, we consider the modular analysis from the work of Hofens' governments and Kills from CCC 2017. And we show memory-type reduction for two variants with implicit rejections, namely FO and FOM in addition. We could show a memory-type reduction for a variant with explicit rejection as well. The variant was named QFOM part by HHK. Now, we recall the hashed L gamma KKM. There are two versions, namely ECIES and the Kramer-Schub version. There is only one, but very crucial difference among them. Anyway, the setup is the following. Capital G is a cyclic group of order Q, where Q is a prime, and small g is the generator of the group. The public key is g to the power x, a random element from capital G, whereas the secret key is the small x. In this paper, we include the generator g as a part of the public key as well. The encapsulation algorithm takes g and g to the power x, samples random y from zq star, and computes g to the power y and g to the power x y. It then derives the key K by hashing g to the power y and g to the power x y. It outputs g to the power y as hypertext, and k as the key. The encapsulation algorithm takes a hypertext y with the secret key x. It computes y to the power x and derives the key by H of y and y to the power x. In the ECIES version, everything except the key derivation is 10. The key derivation is ECIES. However, hashes only the y to the power x, which is actually g to the power x y, instead of hashing both g to the power y and g to the power x y. As we shall see, this seemingly small difference makes a huge impact. So first, we are going to discuss about the Kramer-Schul variant. In this variant recall, the key derivation function hashes both g to the power y and g to the power x y. In the usual reduction, the random oracle transcripts are saved in a table. However, this cannot be done in a memory-type case. The challenge here is to simulate the random oracle in the memory-efficient way, maintaining consistency between the simulation output of the hash oracle and the decapsulation oracle. We need to be consistent with the output of these two oracles by maintaining a small state. The question is, can we apply the PRF trick? The first attempt would be to replace the hash function h. By the PRF, everywhere it is used. Namely, h of g to the power y and g to the power x y can be replaced by PRF of k evaluated on g to the power y and g to the power x y. However, this is not good. The reduction cannot simulate the decapsulation query as the reduction cannot find g to the power x y from g to the power y. How about dropping g to the power x y from the hash evaluation? This certainly helped in the decapsulation simulation. However, the adversary can indeed find collision from the hash query. Hence, the hash oracle simulation is not right. However, we note that things would have worked if the adversary was restricted to making only correctly derived hash query. Namely, it could have worried with only g to the power y and g to the power x y. Unfortunately, we cannot restrict the adversary to make only such queries. So we do the next best thing. We call that in the gravity fulfillment game, the reduction is given a DDH oracle. Using that oracle, the reduction can check the well-formedness of the hash queries. The idea is to puncture the PRF on the well-formed points. That is, if the DDH oracle returns one on input g to the power x and y and z, where y z are the inputs of the hash query, then we are going to puncture the PRF on those points. Now on those punctured points, during the hash oracle simulation g to the power x y is dropped and is replaced by a fixed element of the group. For example, g, the rest of the points are untouched. So from the decapsulation query simulation, the reduction can easily compute the PRF on the punctured points and it needs to compute the PRF only on the punctured points. So this solves our problem. We can simulate the RO for hash Del Gamal Kramer-Schrupp version in the memory type way. This idea extends directly to the module U of the Fusac-Yokamoto transformation. Here, one constructs in CCSEcured KEEM from an encryption scheme, which is one way against plain text checking attack. The same idea works here. Instead of the DDH oracle, we have the plain text checking oracle. Thus, for the hash query, we can puncture the points m prime c prime, where c prime is a valid encryption of m prime. Similar to before, we can replace the message m prime by a fixed message on the punctured points. The decapsulation works perfectly. However, note that the whole thing works because the adversary cannot compute the PRF on this k00 to the power mu and c through hash queries for invalid cybertexts. The generalization to U part is simpler. Here, the reduction has a cybertext verification oracle. The hash simulation is same as before. However, for the decapsulation simulation, we can use the cybertext verification oracle and return a part in case the input is an invalid cybertext. For UM, the underlying encryption is deterministic. Thus, there is no need to puncture the PRF. We can simulate the hash of m as PRF evaluated on the encryption of m. The decapsulation simulation simply runs the PRF. Note, like before, it is important that the adversary cannot evaluate the PRF on the input of its choice. Otherwise, the whole thing wouldn't have worked. The same idea works for UMPARP as well, where the verification oracle CVO is also available. Finally, we come to the ECIS version of hashed Elgama. Here, unfortunately, none of the previous idea works. The reduction simply cannot get g to the power y from g to the power xy. So, none of the other trick of math and PRF idea works here. The challenge here is to simulate or produce h of g to the power xy in the decapsulation oracle simulation knowing only g to the power y. However, we observe that we just need to produce the same result. And it is not needed for the reduction to follow the same procedure as in the decapsulation oracle simulation as it did during the hash oracle simulation. For this, appearing fits the bill perfectly. We simulate h of g to the power xy by evaluating the PRF on e of g to the power xy. And during the decapsulation simulation, the reduction can retrieve the same value from evaluating e on g to the power x and g to the power y. This way, we can make the ECIS reduction memory data when the group admits bearing. So finally, we come to the big picture, the fully secure converter transformations. We note that in the modular analysis to secure converter transformations, there is a pre-processing transformation as well, namely t, which fixes the random coin used in an encryption scheme by evaluating the message m by a random order. The in-CPA to one-way PCA reduction translates via VRF simulation perfectly. This gives memory type reduction for two variant, namely F4 and FOM, where there are implicit rejections. However, we see that the transformations U-PARP and U-M-PARP require verification oracle. In the paper, we introduced another module called V for which we could show a memory type reduction from one-way PCA to one-way PCVA security. Fine, so to conclude, we introduce the mapped and PRF approach in simulating random oracle in memory type setting. This technique is quite simple and it seems natural for key encapsulation mechanism proofs. Interestingly, as we show in the paper, it can also handle correctness errors. Many open problem remains. The foremost in my opinion is about the case of ECIS where on groups when no pairing is available. In a recent result, there has been an interesting approach by Ghushal and Tesaro who showed impossibility result for straight-line programs in the generic group model. However, in the general case, where the reduction can rewind the adversary or the memory of the reduction can actually depend on the memory of the adversary, the cases are still open. Similar open problem exists for the modules FOM and FM-PARP and FO-PARP. Thank you.


Well, as the snow continues to pile up, residents in the city of Bemidji have been unable to keep up with the demand of keeping sidewalks clear. During Monday's City Council Work Session, Council members discussed options for future snow removal. Currently, the city plows only eight out of the 57 miles of sidewalks that's adjacent to city property. While no changes are expected to happen this year, the Council agreed to explore options such as contracting for snow removal on some or all of the city's sidewalks. Even to try to find a place to put all of our public snow as we scoop it up, et cetera, it's mammoth. So there are issues, and you can't, other than finding people, you can't really go there and force people. Currently, if the city receives a complaint about a non-shoveled sidewalk, a notice is issued to the property owner, and if after that the sidewalk is not cleared, a fine could be issued. If you enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland PBS.


On boss talk one-on-one. Yeah, we gon' talk. We gon' have fun. We be on fire. We be lily.  It's a unique hustle nigga.  Check it, check it, check it. It's a unique hustle It's your boy E.C.O. I know more than Moses by the way, and we creating content every damn day Say man, it's going down man. It's really going down man. My boy, my boy He made it man. Hey, he back on the show again, man It's my boy, Smoothie, Poppy is in the building. Oh, Katie is man. Say man, long time coming baby Hey man, one thing I can say man, every time I call you man, you always show up man You ain't, hey, you always there for me man. Thanks for counseling me nigga.  Today is the day that P.M.C. died I believe it in 2007. I want to say man R.P. to P.M.C. man, you know me and you both, you know, we serious about the P.M.C. thing Yeah, so, you know, I see a lot of people posting bun posts to he's and Leo posted My boy Steve, Steve B Lo nobody had interviewed him. If you watching my show, you'll see gotta go back and tap in Yeah, Steve B Lo was a protégé. He has bun mentioned him on Beehive. I brought him over. Yeah I said I watched the interview. Yeah with bun and Beehive. You should tell him I'm the new, I'm the new pro Hey, man Say man, that's dope man, but I'm telling you man Just watching your movement man. All the stuff that you did within the last year, man It's crazy man.  I mean you can listen that thing from start to finish, bro You know, I'm saying so when you done that, I mean, can you top that this year this to 2022? What are you gonna do?   Check out.  Yeah, I started seeing that thing trending in 2010. We were a viral before you could go viral Wow, man, so so what? What what what make what what make you? When you when you think about just the rap game man, what keep you going man? Shit my kids and my family. That's it. That's real.  It will be you from Texas.  I never see this nigga.  You gotta create it Like you have to create it.  You might see me going. I'm talking about the red flag. She yeah, you know I'm saying it to anything to give the folks content of the civil type of person.  So and J. That was just we make them bitches from scratch. They are fresh So it's our first car.   So he made this song me right do somebody else He might like you might be the first one to actually do that, right? Yeah, I have made a song for like for certain people you feel me I made a song for certain people, but that's a that's a new that's a new lane right there I can jump on live and I what type of song you what's up, which I want you feel me a lot of the topic Yeah, yeah certain top didn't go with it Yeah, yeah, so so what do you think man? Who I mean, I've been trying to my best to keep East Texas going man And and just anybody, you know, I know Smitty doing her thing down there anybody knew down there That's that you know of this. There's really making waves that wasn't making waves when we first had our conversation I'm still about the same thing. Yeah, I'm putting you on the spot kind of you know I'm I'm gonna test the counter baby. So I already know that man.  He might be like 16 maybe 15 He young but they own it. They own it. They make drill music already.   B butler, B-U-T-L-A-A man.  Hey boss top 101 you came up on the shoulder today. Hey, Smitty put you out there bad man It's going down, man. Uh, Ben Jones, Ben Jones and Tessa County. I got a song with me.  I got a song with Ben Jones You're bad motherfucker. Hey young young young like that cuz them young boy need that they need that man They need that push man. They need that you bet you been doing it You've been in the game for a minute, man And you got to do something to try to reach back and help the other one So I like that man you coming with a different style I don't have something to get on here and freeze at those kind of question They try to keep it all to they sell Like the way you gonna help and the way you gonna shine help my other people shine down you know though Right once you shine they gonna put your hey this man help me on this man help me on like everybody saying you help me on They're gonna look at you. Yo, so why not keep doing it? Yeah, man And and I got to bring up another subject I've been bringing up every interview cuz dolphin in a lot to the whole and Some people would if you don't want to speak on you don't have to and and really don't have to be a direct speak But but I'm asking you like how do you move now? I know you always move with with precision But how do you move with the way they targeted? It seems like rappers are targeted It seemed like what you're saying is coming becoming your yo yo your reality out of town Yeah, so I mean what are you doing? How you how does that affect you and and how do you look at it? When something like that happened like that and I'm not just talking about him I'm talking about King Von. I'm talking about all the ones, you know, it was this duck that the duckie P. Dude, right? It's I've been a bunch of people man. Just how how does that how does that affect you and what does it do?  Yeah, you got to look at it as a listen I and like you say I I've been moving like that. I already you already moving different you feel me So it's just like being small. I don't never think So you choose nothing can happen to you. Yeah, it can I don't have to happen to partners of mine That's what made me move like they're like Rest in peace call old clock my cousin got killed. Yeah, somebody. I'm wheat. This is my cousin.  He was at all the gas station across the street on the other side of state line You know so, you know, that's seeing it's that she can't happen from that day forward it like man I'm not taking no chances. Oh, no, I'm good.  I ain't got too much I'm I feel I ain't never good. No, no, you you 100% right man did to take those precautions man What talk to me about hell now, man.  I mean, you know Your energy be on just on point bro I love the energy I love the vibe and a lot of people like I say they be tapping in which I seen you down there with Shun cotton, you know shot out to shun cotton you you bless the places I be watching and I seen you In a different I be seeing you moving around bro, right? And I think that's important to you be an H town You on the you you performing on stages in H town. I'm like, okay I see him over there now So I pay I pay close attention to you because I love you you family and and I'm like watching to see where okay What's going on if you need me? I always tell you I'm here But I was at the end of the day. I love what you're doing man. So so Rumor had I got to bring this up at one point But rumor had it that you was signing the 15 oh one did that happen or did you did you change it up? Are y'all still strategizing?   I think all that becomes a factor, you know what I mean I see you and Sergeant J still locking in man. Oh, that's a that's your brother from another mother Yeah, that's like that might hear me. He's brother. That's all I can be man. Shout out to Sergeant B man, so I can be an engineering beast man You feel me? I saw Sergeant be my voice and so I think my heartbeat. Yeah, I'm just like like I know coming from out East Texas man That sound the way that he a lot of the reason the engineering that Sergeant B Do there's a lot of reason that a lot of music sounded the way it did and gave a lot of To the whole East Texas movement because really if you really look at it, it takes it really ain't up to par for music and resources and shit correct Sergeant B is really a gay keeper down there for us keeping the sound Up there with everybody else in the major city correct. I see that you don't really get no fall off No, no, no, I see that and I think that's dope Shout out to Sergeant B pose be coming back on the show sergeant. I've been hollering at you man. What's up, nigga? Hey, you're boy EC. Oh, you know how I call But at the end of the day man, so what can we expect though? Like like how you go is it is? Can you name anything that's coming in 2022 for a project this account of baby two coming? Okay? Okay, Jay, you know It's gonna have some nook beats or sergeant and nook link up and they go in on beat shot the nook ahead Okay To test the count of baby two and the features Man ain't got no feet. Well D raw. I got D raw. D wrong. I D wrong me Two days ago, you see man, man when I pull up to the studio Jay gonna be me Jay D rock on pull up.  That's all at the studio and pull up man that because you like that work He liked it work man.  That's how you win, right? Right every time. Yeah, so so do you can you probably okay? So y'all send each other be since sergeant Jay is down there when you go to it takes a county I might be Jay and say you I hear Jay and say hey, I'm gonna go to the studio tonight.  It's time to go in there You feel me? I want to be fresh.   I'm going in then I listen to the rest of them. But yeah, that's how it is I seen you did something with it was I think it was two guys from Marshall Did y'all ever even do did that ever even come out? Or did you still did y'all have like the four horsemen or something? It was you are your Me lose say on site Joseph. We say yeah free lose say free lose say man free that boy lose say So I did that that project never came out now. We just we uh, we did two song We got two song put up.   Yeah. Yeah. Yeah, I am part of it Is she he didn't got too big for what he is exactly you I grow where you a being in East Texas It ain't once you get some friction, you know some static going, you know, I'm saying you get rolling in the east Shit, it ain't too much. It ain't too long before you get so you gotta go to move So you agree with Boosie when he say you gotta leave your hometown It ain't just a smallness of it just the fact of how big you getting wherever you at Yeah, is it important or do you feel like you can live in your own town after you success? She I think you gotta move. I don't think nobody can no, I don't think not not in this damn time No, it's like When you out there when you really doing good is back in the day in the 60s and the 70s and all that coming Sam cook was the man You only knew Sam cook was the man we heard on the radio when you went to the show to see him Now if you the man love you got any type of way they are you on Facebook you on Twitter you on Instagram?   Hey, that's the worst one. That's the one Yeah, that's the one that and and folks like that a look for another nigga. They don't like the time Yeah, yeah, yeah, yeah, I recruit niggas recruit now. They recruit niggas.  He'll wait till you get what you feel comfortable when you feel like okay I heard he on a really careful. You know, I'm saying they take team up to try to figure out a way to get you out of here they built But when you really when you succeed it's a I say the worst thing you could do is have a rich out Wow, I heard that too.  So when when you look at boss talk 101 what a boss is talk You've been checking the side you've been seeing the move man.  They had to come down here and and and we end up being bland We end up in worlds. Do we in every street? Global global media Everybody man. He boy. Hey, it's church. It's everywhere. I know what a heaven I said, God, yeah, but I'm talking everywhere. I saw money the conch on those e-mail man You posed to man. It's all good. Come on, man But you know, I think a lot of times like when you look at all the different platforms where where they had they You know, they leap they great leap They go through a real moment like that to be honest with y'all remember breakfast club Ray J You know what? I mean, that's how that thing first popped off when he got into it with fabulous and he was calling up down the phone Or that that bird man movement when bird man My name boy He puts the whole thing around in there man too many interview Too many did more numbers in any other view they ever had You see what I'm saying? So I mean and you know, we come in the dough loving and trying to do the right thing Family and bringing in black business ownership and I think a lot of times people see people on the channel And that's why people like you is important for me because I got a level off and balance and say hey, man We ain't just about that man We really showing you what's really going down with what we doing in Texas what we doing in and you know I mean the new style for the new folk The one that be heard. I really know. Yeah. Yeah for the new folks and you gotta realize though We only been in the game for 11 months really when you really calculate it This the level month and he and he's the major one. Yeah. Yeah. So that's a blessing. So at the end of the day What's up with it? He they do y'all still been linking up.   That's like he to like my Hey, they're my cameraman.  He's a cool man. He's always working to you feel me So you see you see that move when I really cuz I you see that move when I brought him up to Jefferson when he Filled Charles didn't it and we were down there to Juneteenth. I seen a piece And we didn't never just get it back get it together but But at the end of the day, it was a good move because you're Juneteenth and we just wanted to do something He said, you know, that's my area down there. Yeah. So I Shot I trill talk no pill talk. That's my nigga right there, but say man. Hey, man Hey, we've been cutting up to I Still talk be rid of the fight by these Hey, that nigga really has fight Yeah, very passionate And he gon fight. Oh, yeah, he gon fight. Don't think that nigga just don't want to talk to you. No, no, no.  Yeah.   It's a thing called respect, right? So we are long for that respect, right? So that's the thing I say, man If you're gonna do it man be tasteful with what you do man when it comes down to You know dealing with different people but at the end of the day people gonna have any differences Some people not gonna I mean because I think a lot of the conversation about these texts It was right a lot of times people don't support they own in these texts But a lot of times I like I told a certain people that was on this panel in Dallas They don't support they own in Houston. They don't so it Where you go, you're gonna give more love from another city Yeah, you're gonna blow for somewhere else for you blow back home Well, look how you did it when you went to school and all that and you you brought all of that Which that's what I like about your brand too.  Yeah, here why being it's gonna leave you come with me. I'm a good person. I don't Yeah, I know good beating. Yeah, if I if I get if I get into with somebody with my fault You feel me because if I feel my fault, I'm a side wrong Man, I'm for the link you with somebody man. Shout out pgf shout man. That's my boy. Come on. He's gonna do a whole A pimpleation type album.  I'm gonna see if you and him can clap He doing the whole he's gonna do a whole thing and he I love the way you move I was just with him the other week him and the rest Okay, yeah, but man, I gotta get y'all together because both of y'all be on that P. M. C You know, I'm sweet zone junior junior Okay So Okay, so what's what's up with who you want to work with this year now?  We ain't man. He's gonna come out.  He's like kind of like you feel me? He see what's going on now. Like he say, okay.  Yeah, so he gonna he fun to come on We I shall hope so man. I've been waiting on that boy and you know, I'll be getting frustrated, you know But it's just like anything else we wanted when we wanted I'm uh, if I go tomorrow, he could keep going with me already. He don't have no choice Yeah, he ain't got no choice now. Like we locked in that like my brother. So we locked in nigga Ain't got no choice man. Like you say you go to the club. We roll the girl the niggie five Yeah, we gotta do it together. We gotta fight. We gotta eat together. Every day, man. Every time, bro So what's the I that takes kind of uh, baby to what what's going what can we expect on that thing?  I'm talking about front of bag personal stories Oh, you know, I don't really hold back no more. You feel me? I'm at a point now Well, if I did somebody did it I didn't see it. I don't hold back on what I'm gonna say Oh, who I'ma mention you feel me deal with it is But it's what we from east Texas. You feel it? Yeah, yeah, yeah It ain't something just I we could it's something we could represent but it's something everybody gonna feel Okay, tell me about this mic hanging out the air man on this mirror want to tell me What mirror on the thoughts? What is this mic hanging that I see everybody doing man? Oh, yeah, that's the It's the mic hanging out the air. It's the mic hanging out the air movement I don't know where it came from. I know visuals by D. Have been doing He had been there. Yeah. Yeah, he came Shout out to visuals by D. Shout out to heat hd. Oh I don't know where it come from, but I love it. I love seeing it. It's all welcome to the yeah It's all this is extra content for somebody You feel and that's why I sit here.  I was shooting the video.  So when I shoot a video, I'm doing a future for somebody or We shoot the video for a future out of done Man, he the pulling up to go. So we leave and he like she you want to go do One of them things. Yeah, what am I saying here?  So uh, that's the kind of stuff I like I love the way the internet move man. Like you never know what's gonna be next man So are you trying to break into tiktok or any of that man? Man, I'm I'm trying I don't make tiktok music. Yeah You don't know You don't know what's gonna happen. You don't know.  Yeah. Yeah, she goes to the same thing. He don't like uh, he not a tiktok rapper But he should go on tiktok though. Yeah That's where it broke. Yes. Well a lot of it broke on that.   Uh, uh, I'm on IG most of most of the day Man 70% of that I'm looking at networking. I'm talking.  Right So you get on there like damn ain't nobody really rocking with the reds Shout out to the reds. He on there. He just many of them talking about that down there He said he had to get on there man.  Like the way that you you don't care. I know no hold no balls. Hell, man I don't have a fun with this shit man. Yeah. Yeah. Yeah, I'm having fun with it So I see and I don't know if you want me to bring this up I see you're on online with the baby man the way the way I say man, how that make you feel man All right, man. I got my running back already. I love music, but you know how I feel about football, man I got my running back. Yeah How's it dealing with that little man? He growing man. He's growing man. He's just like me. I see he's just like me. He reminds me of me.  Yeah, I get to go home and look at him I'm like, okay. Yeah, I need to go. I need to get to work because I take miss it I I take missing out on a month or two.  That was funny to me when I made that nigga say smoothy I pulled it out of you. Yeah, come on You make you make them all you make them tell the truth.  Yeah, but he was he was so he just telling the way he is here You know, everybody can have their opinion man. So what do you think about these lists they be doing man? Yeah, you be on a lot of them lists too, man. So So do you think it's biased or do you think the list is something that's needed or I mean, it always be biased to the person that's making the list. That's that's that's that's it. That's it.    That's it but when uh A blog or anybody said make a listen blog you you ain't good enough to make titanium Nigga cry about it all day won't maybe we'll make stated man show I'm I'm I'm the type of nigga. I'm bill. I'ma show you or I can't do it Oh, let me show you this Same same same approach. I took the football. Like, uh, you come from Tessa County. I'm a I come in one man as a freshman. I ain't Are you not gonna play we're gonna research. All right, let me show you some I'm for the key You're number one defense. I'm scouting him out. All right, let us we're gonna.  Yeah, that's what I thought. I know it You messed up.  So it makes you motivated.   He don't know.  Man, I'll just be on one lie.   But yeah, I'm gonna got to though when I I'm laughing And the reason you got to it because that's I mean I mean you think about the ads are being paid through ads paid through live Reels and stuff those opportunities that making finance because we already know the game How tough it is in the music industry right now to make money off of the music because of the way people doing it So you got to come up with strategic ways Like I said last time merch and all that is so important because you guys are working so hard And doing the music and putting so much into a time in elford We got to come up with ways to turn everything ROI and you know what I mean get it popping, man So where you at where you at tonight? What's going on tonight? You chilling or you going you going to be in the studio? I'm with 15 on one tonight. Oh, yeah, I'm with 15 on one tonight. We stepping out. She y'all stepping Yeah, you know, I just in the eights. I came back a day early. I left a day early Then they came back. Oh, they came the next day. So she we are she They said they were going to some club. What did they say they were going? You remember I forgot. I don't even know the name. I forgot the name of the club.  So yeah My co-host gonna know now we do got a photo shoot tonight Hopefully we can rock that thing out and and and boss talk one on one gonna be everywhere everywhere Yeah, yeah, yeah, yeah, we it's time It already is I know it man. I thought I was going to drop the merchandise as soon as he We already did. Yeah, I'm killing them with the with the the outfits. They've been selling like crazy Yeah, yeah, my wife man. Shout out to mr. Mako, man. Shout out to mr. Mako. Man, she put it down. I owe them all G that man, he keep out digging right No, man, we a we got something for you. She sent you something. She always gonna look out man The creamer came really So yeah, so man, I just you know, like I said, man, I thank god that our relationship man is real It is surreal that is that we able to you know, boss talk one on one you family And anytime you anything you doing, man, you can tell me I'm pulling up. You call me. I'm coming, but He can be an ATL or whatever. He was somewhere else. I don't care. He could be in new york, nigga I pull up. Come on. You know what I'm saying? I promise. I believe y'all I promise y'all pull up nigga, and I'll be going places just be on the boss talk thing But if I need to switch lanes and go for something else, then I don't don't trip Yeah, you know, I'm saying a lot of people, you know, they tune in the boss talk They don't really know what's going on with the way we moving down here in Texas We trying to wake these boys up, right? That's the way Man, listen, man, uh, like I said when I came into the game, I didn't even know that we were for the rock out Like this, uh, I knew it probably was gonna do it though Remember we he said it from day one when we drug the table up in there And he said man, this thing's gonna go man. It's over man.  Hey, man, so how can people get a hold to you, man?  Instagram Twitter Facebook. Okay. I'm gonna ask you your top three artists all the time They might change from last time because it things tend to change if you had to pick three artists that you Would say is the top three artists all the time. Who would that be? Number one Boosie Y'all agree with that Boosie number two. Number two.  I knew I was gonna get in there. Yeah, nigga. Yeah, nigga. That's the reason there is a boosie It wouldn't be a boosie without pmc. I'm gonna be me without boosie. Hey I'll say boosie because I'm from the south. Yeah. Yeah. Yeah.  They love it. Boosie ain't gotta We love him. Yeah, boosie got a performance song after 2012 You know, uh, the message just dropped. I ain't listen to a g I'm I'm waiting till I go out of town. I'm sorry to Ben Roach. Boy. He went back to Ben Roach It's a slap. I seen young boy post it. He went he got a song called uh free little top for real He rocked with he rocked with your boy. He love him. He got a whole song about Ben Roach Like the whole song about Ben Roach. It's old man Man, that's that's what it be man. When you really when it's really going down, man The boy like say he wild with it too. He on play when you he just like you Meet him. He's just loud with it. I got boosted tattoo, man. For real? I got boosted on my own boosted tattoo on my own I got boosted like wow Yeah, I know you had a film. All right, so you don't play with it. Yeah, man. I I have to south When you see boosie getting in trouble to saying all the stuff that he's saying Well, I say get in trouble because you really can't get in trouble when you When you when you free like he is He can say whatever you want to say Even if they try to eat they can cancel boosie the day to cancel them. They didn't count it on instagram They didn't cancel them three time four time. He still come back boosie get a show the next He got showed three days later. They don't kill the south don't care about that No, man, we got love for boosie, man All the way niggas add straight out of Dallas all the way all the way to houston back down Four month back down the street boy. Hold up niggas all the way back down and test the count of niggas And I'll never forgive them for what they did to Clyden Ray, man.  Oh, these two niggas right there.  He hardly sang rap.  You jumped that line. It's right there. It's right. I mean you can't you What about 10 minutes over into it when you end up by the mall and stuff you can be in you can be in Texas And Arkansas at the same time Split the land Yeah, so, uh, man, so what what's the good food place out there? I can eat out there in Texas count. I go home Man, they got off the fire.  They sell everything. You can get your uh, you can get some dressing one day Are you serious soul food? I'm going soul food. I'm going. They got some good. I need one.  It's right. And it's right. Hey matter of fact when you was at the studio that day I came in eating some you did.  I'll do it Playing some playing some of that. Uh, zaza Hey, man, so hey, man, um Anything else you need to get off your chest, man. Thank you so much, man. We love you smoothie, man You my god, man. You know already, man. How we do it, man. Listen, man It's me and you man against the world nigga. It's going down boss talk one-on-one Tied in with smoothie poppy and it's going down, man. It's been another great segment of boss talk one-on-one, man


Hello, I am Ick and today we will look at the water holding capacity of your soils. The capacity of a soil to retain water will determine what plants and trees can grow there and their growth rates, which is especially important in the hot and dry seasons and in regions where the certification is occurring.  For this method we will need a watering can or a hose and spray nozzle, three or four trash bags, small tarps or pieces of thin sheet plastic, one sensitive scale, preferably 0.01 gram sensitive, a spade, a clean bucket and one clean ziplock bag or a container to hold a soil sample, a kitchen sieve or a screen material with two millimeter holes, a mortar and pestle or another tool to help you grind the soil, a kitchen over or toaster to be set at 105 degrees centigrade, an oven proof dish to hold the soil in the oven. Ideally, we do this test after a substantial rainfall event with clear forecast for the next three days. First, select and mark out three sample sites in the zone that you are restoring. At each sample site, free a one square meter area of soil from vegetation and soak it slowly for several hours until saturated. Then we cover with a plastic sheet and wait for 48 to 72 hours. The sheet will prevent water loss through evaporation. After the two or three days, remove the plastic sheet from each sample site. Then use a sharp spade to cut soil samples. These should be 15 centimeters deep, 5 centimeters thick and the width of the spade. Repeat this for all the sample sites. Then mix the samples together and omit any roots or stones in a bucket. Take about three to six cups of the mixed sample and put it on a pan or another high temperature container. Zero the scale, weight the soil and record its weight. Then bake the soil in an oven at 105 degrees centigrade for 24 hours until the water has evaporated. Then let it cool. Grind the soil sample and sieve it through a two millimeter mesh. Sifting out all the remaining stones and roots. We measure and record the weight of stones and roots. Then we weigh the dried soil sample and record its weight. We then weight the empty clean recipient and record its weight as well. Finally, we're able to calculate the water holding capacity in kilograms of water per kilograms of soil using the water holding capacity equation included in our datasheet. When repeating the test, make sure to use the same procedure in order to get meaningful results. A low water holding capacity could limit soil biological activity and plant growth. But usually, water holding capacity increases with increasing organic matter content. If we repeat the test throughout the restoration process, we're able to see if our restoration efforts are successful.


Today, we're going late Roman and early Christian architecture. This is Constantine up here. But before we talk about that, I just want to go back a little bit to Vitruvius, whom we were discussing earlier in this class. Vitruvius, as we mentioned before, was a Roman architect and architectural writer who came up with his 10 books on architecture, which provided architects with a theoretical basis on how to design, how to make choices, talking about things like the three qualities that all architecture should have, firmness, commodity, and delight, or structural integrity, functionality, and beauty.  What does decorum mean? Tom, what's decorum? Appropriateness, exactly. So it was very important that if you were designing something, you responded with the correct type, or the correct materials, or the correct use of the orders. The practice of architecture in those days was quite different than it is now, where there's a lot more license to experiment with form and material and so forth. We looked at a number of things that Vitruvius recommended that the architect should consider, things like the making of machinery, the making of technological devices like the water screw for drilling a well, or things like the clock, or things like the city plan. And I mentioned that all of these tended to have more or less the same diagram, which was radial, a circle with radiating lines. It gives you a town. It also gives you a clock. It also gives you a water screw. And that's amazing how convenient. You don't really have to do much doodling. A circle with radiating lines gives you everything. But probably one reason why Vitruvius favored this diagram is that it was cosmologically significant. The idea of the circle has this pedigree in classical thought. Going back to Plato, going back to Pythagoras actually, before Plato, connecting it to the cosmos. The circle is a diagram that connects it to the cosmos. And Plato, in his book The Timaeus, talks about the four elements, earth, air, fire, and water, and the cosmos. And he attaches a geometrical shape to each of those things. So the one thing that Vitruvius describes that relates to these geometrical figures, the circle, and also the square, he talks about very, very specifically in his treatise. And I'd like to go over a bit, because I think it's a really interesting question. And there are some interesting solutions that came up. And that is the idea that the human figure somehow corresponds to geometry. That's fairly amazing. Because classical thought tells you that there is a kind of dissonance between things in this world, things trapped in time and material that corrupt and rot and pure ideas. But there is also this idea of microcosm, that here on earth there are little images of the world that we can understand, a little cosmos, microcosm, that reflect the order of the divine. And so it's of interest to try to see and understand and even make models of the microcosm as often as possible. So when Vitruvius gives his description, and Vitruvius doesn't have any illustrations in his text, the test from antiquity is completely literary. But he describes the correspondence of the human figure to these two ideal shapes, the circle and the square. On the screen we have Leonardo da Vinci's drawing from around 1500. But before we talk about this one, I want to show you some other attempts. Because it's one thing to say circle and square, guy in the middle. It's quite a different thing to try to solve that problem. So you have a number of these perfectly smart, perfectly talented Renaissance guys who are really interested in recovering the thought of classical antiquity. And they're happy to have Vitruvius because this is a window, a specific address about how the ancients thought. And Francesco Giorgi draws this guy, which I think it's not the best attempt when you say this kind of floppy man standing in the circle. And the square is really not addressed at all. The square just happens to be there as something that the circle sits in, but the human figure is not really generative of the square. And it's barely generative or generated by the circle. So you get another guy, Fra Giocondo. And he's really trying hard to figure out what does Vitruvius mean? How do the circle and the square correspond? What is the logic between them? How is the human figure somehow implicated? And he says, well, okay, this guy can kind of give me the circle. And this guy, and look how hard he's trying. Look at that sad expression on his face. He's giving you the square, but he's about to explode in the task of giving you the square. But in the project of Fra Giocondo, the circle and the square don't come together. This apparently easy thing that Vitruvius describes is a hard thing to accomplish. One of my favorite responses to this problem is Cesareano. And Cesareano is really a friend of Leonardo da Vinci. He's a guy up in Milan, smart guy. And he's pulling out all the stops. Notice how people like Fra Giocondo or Francesco Giorgi are just using the big geometrical figure, the circle and the square to try to solve the problem. And Cesareano's a smart guy. He's saying, what about the grid? What about the idea that within the square there are modules and subdivisions that can begin to help me organize this thing? So he's got the big gridded kind of graph paper thing. He's got the circle, he's got the square. There's this kind of figure ground lock between the circle and the square. And this guy with size 16 feet and lobster claw hands is stretching out and he's kind of doing it.  There's no confusion about that. People don't exactly look like that. But Cesareano still has another conundrum here. And that is the center. What's the center of the circle and what's the center of the square? And how does the human figure relate to that? So classical thought will tell you that the navel, the omfalos, is the place where the soul resides. And so if there is something spiritual about man, then the navel would be the center of that. The navel would be the center of the circle, this cosmological diagram. The square, according to classical thought, according to Plato, is the figure that corresponds to the earth, to the mundane, to things that are trapped in time and material and heavy and ponderous and animalistic. And so what is the center of the square? And Cesareano thinks, hmm, maybe the reproductive organs. So if I gave you a less pixelated image of this, you would be blushing and you would be shocked because Cesareano's little guy is desperately trying to have both circles aligned and his little man is extremely excited. This is one of my favorite Vitruvian men. This is by Francesco di Giorgio. And this guy is just chilling. Like, I'm never gonna get there. I'm just gonna relax and just kind of hang out there. And it really takes somebody of the caliber of Leonardo da Vinci to figure out a good way to solve the problem of Vitruvian man. And one thing that Leonardo does that the others didn't think to do is simply not make the circle and the square concentric. Over here we have Cesareano doing what seems obvious. If you read the literary description of the Vitruvian man, the circle and the square are concentric. They have basically the same kind of ideality and the diagram that Cesareano comes up with is pretty easy and pretty straightforward. But Leonardo slips them so that they rest on the same ground plane, but the centers slip with respect to one another so that there is the orthogonally disposed figure of the person that is descriptive of the square and described by the square and there's the radially disposed person who's descriptive of the circle and described by the circle. And the way Leonardo draws this guy is not simply a description of human anatomy, but rather, if you look closely, you'll see that he's really parceling out the figure into these interrelated pieces. There's this kind of geometrical diagram showing you corresponding measures. And there's also this idea of the slipping of the two centers. So for Leonardo, the center of the circle is the navel, the spiritual center of man, and the center of the square are the reproductive organs and that works out incredibly perfectly because the figure seems like a person and not like a monster or a lobster man and the two meanings are called out. Leonardo is mathematicizing the human figure and at the very bottom of his drawing, there is this linear clarification of what the measurements are and what Leonardo was showing you is that he's sort of unfolding a golden rectangle spiral and there's something about the human figure and these ideal geometries that correspond. And just as an aside note, Leonardo's writing looks crazy and you might think that this image is reversed, but Leonardo was always afraid that people were going to steal his ideas so he simply wrote backwards. He wrote everything in mirror handwriting. So if you could read Renaissance Italian and you had a mirror, you'd be in luck here. And sometimes Leonardo had so many ideas, he would simply take a pen in each hand and fill up two notebooks with two different sets of ideas at the same time. Another little bit of backtracking, not necessarily backtracking, but clarification I want to go through is this whole idea of the Roman Basilica and we talked about the Roman Basilica when we were just discussing the fora and it's something I just want you to understand as a very important type as we go forward in architectural history. We saw it over here, this is the Roman Forum and we see Basilicas lining either side of it and the purpose of the Basilica is meeting all. The purpose of the Basilica is not a religious building at all. The religious buildings were temples like little temples over here and the way in which the temple operated did not involve a congregation. You didn't have a group of people assemble inside a temple. Rather, the priests would perform the rituals there and the observers of the rituals would be outside. But here in the case of the Roman Basilica, this is a public meeting hall and its purpose is for the assemblage of large crowds. So this is essentially the type of the Basilica and notice that it has aisles along the side and the large space in the middle is called the nave. Perhaps you know the word navy, is this a familiar word to you? Naval battle, all that has to do with ships. Nave is the Italian word for ship and people who built the wooden trusses for churches in the early days were often the same people who built the wooden structure of ships. You just turn them upside down. That was certainly the case in Venice where the carpenters at the arsenal would build both ships and truss work for churches. This large space in the middle is the nave. The space on either side is called the aisle and light comes in from above through clear story windows. So that's basically the type. This is the Basilica of Maxentius and Constantine. Very late, 308, 313 BC. This is a heavy masonry building. These are arches. These are large vaults over here. Typically it's a columnar structure. This is simply a transformation of the type very slightly and this is where it's located. The Basilica of Maxentius and Constantine right here at the edge of the Roman Forum. It's probably surprising that there would be a Basilica for both Maxentius and Constantine at a moment Constantine and Maxentius were joint emperors and then they each appointed a kind of helper buddy, a Caesar to help administer the empire. So you had four people ruling in tandem. So the empire begins to splinter in administrative sense as well as in a political and militaristic sense. By the time we get to 300 AD, the stability of the empire is falling and it's being threatened by a number of things. One thing that it's being threatened by is the incursion of barbarians coming in from the north and attacking people. Another thing that it's being threatened by probably is one of the glories of Rome, the plumbing system that brought them fresh water, that brought them toilets and plumbing. Also used a lot of lead and the lead poisoned people and began to kind of diminish their capacities. But also from the glory days of the founding of the empire with Augustus back in 27 BCE, there become increasing schisms and the empire begins to fragment. First into co-emperors who, here's just a diagram showing you some of the fragmenting of the Roman empire. So we have 27 BCE where Caesar Augustus consolidates Rome and declares himself to be the emperor. 250 years later, Diocletian divides the Roman empire and you have an Eastern empire and a Western empire. And Diocletian is probably the toughest on Christians. He is very interested in suppressing this new sect. In 330, Constantine transfers the capital from Rome to Constantinople. Rome remains the capital of the Western empire. Constantinople, present day Istanbul, becomes the capital of the Eastern empire. And the Eastern empire grows while the Western empire shrinks. And the reason the Western empire shrinks is that the barbarian invasions are coming down through Italy and Constantinople is more or less protected from that. And this diagram just shows you some of the incursions of barbarians that had to be dealt with and that began to really attack Rome in a big way. While Constantinople over here, headquarters of the Eastern empire remains relatively at peace. Hence explaining the difference in prosperity. And Rome's a good target because Rome is rich, right? And if you can get all the treasure of Rome, you might think you would get the kind of lifestyle that the Romans had.  And pretty much by 410, the empire in Rome collapses and ceases to be a viable state. So the empire begins to splinter in administrative sense as well as in a political and militaristic sense. Different emperors, different co-emperors are mounting their own armies, levying their own taxes and the whole value of the Roman coinage begins to diminish and this makes it harder to enlist a good army to fight. Another factor, and this is at the same period that Augustus becomes emperor and declares eternal and inevitable right of Rome to rule forever, you have the rise of Christianity. If you look at the dates of Caesar Augustus, it's right at the time of Christ. And so you have religious splintering going on in ancient Rome also. And Christianity was certainly not the only new religion that flourished. There were a lot of variations of religions that in order to keep the power of the state intact, it was convenient to persecute all of them. These are just some images of the suppression of Christians during the time of the early decades of the new millennium or the 18th. This is St. Peter, one of the apostles, and he was crucified in Rome and they crucified him upside down. So not only did they use this tool, this symbol of the state of Rome to crucify him, but they did it upside down to underscore the subversiveness of his teachings. And so because of that, the early Christian church became a covert institution. It became a kind of secret hidden institution. These are some catacombs over here. And people would hide. There would be no desire in the earliest years of the Christian church to make visible the presence of the church. Rather, you would go into the catacombs and secret meetings, or you would simply meet in people's houses. So the idea that the church would become a type, the Christian church would become a type, didn't establish itself until quite late. This fragmentation of the Roman Empire into first two and then four goes by various names. When it is the ruler of four, it's called the tetrarchy. And that was established by Diocletian in 293. And this was done really as a way of spreading out control so that the barbarians and the collapse of the economy and various other problems would be dealt with. Constantine is the one who essentially divides the empire between the East and West. Constantine goes to the East, Licenius goes to the West. And the other two are gone. Constantine founds a new city when he goes off to the Eastern Empire. And he wants to call this new city, Rome, which is a catchy name. But people so honor Constantine that they call it Constantinople. This is just Diocletian's palace. We looked at that before to give you a sense of what the architecture was looking like at that late moment. But the architecture was also looking increasingly complex. And already when we looked at the Villa of Hadrian in Tivoli, we saw the simple types of temple or solos becoming intricate, becoming implicated by inner penetrations of other volumes and other objects. And that certainly becomes the case as we move forward in the Roman Empire. This is the church of Minerva Medica from 330 AD. So the empire is already dividing by this point. And it's a nice elaboration on the kind of pantheonic type. It's an elaboration insofar as the perimeter of the building begins to express the interior. So you get this kind of wonderful, crenellated surface. It's a centralized plan, but it's a centralized plan that begins to become capable of spawning secondary, centric organizations. So this plan down here shows you how there would have been a little antechamber or narfex that you move through. And there would have been these cells on the side with support buildings, so that it really looks almost like, I don't know, some kind of organism that can keep spawning and creating secondary organizations. Much, much more complex and also much more structurally invented. If you look at some far flung late Roman churches like St. Gherion in Cologne, Germany, it's kind of a transformation of the Minerva Medica type. You've got this wonderful crenellated surface in folded surface that folds up and becomes a dome.  And that's the idea of center that we see so clearly expressed in Minerva Medica begins to slip in favor of a description of axis. So formally there's a nice manipulation from one program to another. And possibly one reason that these programs are changing already in 380 is that the state religion had shifted, which is San Lorenzo in Milan. Another super hybrid centric plan where we get these kind of concave, convex, infolding shapes like the ones that we saw at Hadrian's Villa but now amplified by the addition of these other kind of weird ancillary centric organizations. And probably one reason that the church plans begin to change, particularly the church plans begin to change in favor of the axis has to do with Constantine. Constantine, as we said, was co-emperor with Maxentius. And there is a battle where Constantine is trying to defend Rome and his armies are moving to the north of Rome and the invaders are coming down toward them. And Constantine, this pagan general, this pagan emperor, sees a cross according to the legend floating in the sky. And he says to his armies, go, follow the cross, take out your swords and follow the cross. So Constantine's army, following this glowing cross in the sky, defeats the army of Maxentius and defeats all the enemies on this bridge called the Milvian Bridge. And at that point, Constantine decides to legalize Christianity. Constantine doesn't actually become a Christian but he legalizes Christianity so that instead of becoming this secret sect meeting in the catacombs or meeting in people's houses, they become a more public institute. In fact, a few years later, the emperor Theodoric would actually persecute people for not being Christians. So there was quite a transformation at that point. And so when we look at things like Saint Geryon, Heilege Geryon, the idea that there's an axis that begins to transform this centric Roman type has a lot to do with the changing idea of liturgy that goes on with Christian churches. That the church is now congregational so you want a large interior space and you want to have a kind of hierarchy between where the congregation are and where the altar is and the priests are. And so the form of the building stretches to accommodate that. Notice that this terminal lobe in the plan becomes larger, becomes a place to put the altar, and also processions are possible. So the dopey center that we have over here, which is actually a beautiful church in Minerva Medica, now becomes something typologically much more capable of accommodating the sorts of services that the new Christian church would have. This is not really what Constantine looks like. He's not a ghost man with a head and an arm, but there are these fragments of statues of Constantine that remain and they are colossal things. This is a reconstruction of the pieces that we have and what they would look like because Constantine was very successful militaristically, not only did he defeat the co-emperors, Maxentius and Licinius, but he also fought against the Franks, the Visigoths, the Samarians and his reign and even was able briefly to expand the Roman Empire which had been contracting like crazy up until that point. So these are the colossal heads of Constantine that we saw reconstructed in that image, colossal hand of Constantine, his magnificent feet, and here's a person to give you a sense of how big these elements really are. They really are these gigantic things. So Constantine was a good Roman in that he built giant statues of himself and built a triumphal arch to celebrate his victories. This is the Arch of Constantine, probably the most spectacular of the remaining triumphal archers in Rome, but he also began to affect policies that changed the attitude of the Roman state, particularly toward the Christians. In 313, he signed the Edict of Milan and the Edict of Milan declared that all religions in the Roman Empire were fine, that there was no dominant religion. All religions would be tolerated. And then a bit later, the Council of Nicaea was established to clarify the teachings of the Christian church. But as you know, the Nicene Creed, it was formalized at the Council of Nicaea back in 325. So the Council of Nicaea also begins to establish what counts as orthodoxy and what counts as heresy. And at this point, there were all kinds of branches of Christianity, many of which were stamped out not sufficiently orthodox. Well, Constantine probably never converted Christianity, or if he did, it was a deathbed conversion. As he was dying, some stories say, he said, yeah, okay, Christianity, why not? But his daughter, Constancia, was a Christian. And when she died, she was buried in this little martyrium. Martyrium is a type, and it is a tomb for a martyr, really. And it's typically a centralized building. And the Santa Costanza establishes the type. And as we move forward, you see variations on the theme of the type. From the exterior, this big round lump of a thing, kind of looks like a mausoleum that you might have in Roman times. But a Roman mausoleum wouldn't really have interior space. You would simply stick your dead emperor in there, and that would be it. But part of the idea of the martyrium is you venerate the martyr. There will be relics of the martyr. Do you know what relics are? Like an ear, or a nose, or a foot, stuff like that. Notice when we look at the plan of Santa Costanza, it's almost a pholos inside out, right? Remember the Greek and Roman type of pholos, where you have the cell on the interior wrapped by the columns. Here we have the solid perimeter with columns on the inside. Or you might say it is like an onion. You're slicing through a series of concentric rings so that it's possible to have an experience of it that involves the exterior, but the real experience would be the interior. And if you look at the section, the section of Santa Costanza is the section of a basilica. But instead of being nave aisle aisle that gets extruded along the line, it is the notion of the ambulatory that's this little circular aisle that gets bent around a center. And it's little, but it's beautiful. And there are all kinds of interesting things going on. If you look at the columns, these are Roman columns. And some of the columns that you find in Santa Costanza and you increasingly find in early Christian and medieval architecture are just spoils. You just take them from a Roman building and you put them on there. And so there might not be the same attention to the integrity of the orders that you would have had in Roman architecture. You just borrow what you can. And there's a kind of double value in this appropriation of spoils from another culture. In one sense, we don't have to carve a new column, but in another sense, you are demonstrating the eclipse of the old order and the ascendance of the new order by reincorporating pagan elements in a Christian church. And after Christianity becomes legalized after the edict of Milan, there becomes a project to build some Christian churches. The Lateran Palace is given over as a new seat for the Christian church. And it gets developed into the first early Christian church, St. John Lateran, which I like to call San Giovanni in Laterano, just because it sounds so great. And this is not what St. John Lateran looks like now, but this is probably what it looked like back in the early Christian days in 313 AD when it was constructed. And it really does more or less deploy the type of Roman basilica. We see two aisles flanking either side, and we see a large nave. We see also something that we hadn't seen in the Roman basilicas, and that is this thing over here, this element that crosses perpendicular to the direction of the main nave, and that's called the transept. And the effect of putting the transept, this perpendicular element across the nave is to change this linear type into a cruciform type. And that's kind of fabulous also, because the very plan of the church becomes a symbol of the church. So if you look at the architecture of these early Christian churches, like the church at St. John Lateran, you see something that looks really quite modest compared to things that we were looking at in the Roman work. For example, think of the Basilica of Maxentius with these enormous vaults, these great, great kind of heroic swoops of space. This is a modestly constructed architecture. This is a kind of bearing wall building with simple columns supporting the nave on the inside. Notice that the windows are not large, but they're just punched windows into the wall. Here's another early Christian church just a few years later than St. John Lateran, old St. Peter's. By the way, I wanna say that St. John Lateran became the Cathedral of Rome. And cathedral is not just a word for big church. Cathedral specifically means the place of the cathedral, and the cathedral means the throne of the bishop. So this is the seat of the bishop of Rome is St. John Lateran. And that's not true of St. Peter's. This is where the vicar of the church, the pope has his seat. So this is St. Peter's Basilica. And if we look at the section, this probably gives us a clearer idea what an early Christian church looks like. There's a real thinness to this compared to the heavy masonry that we saw in Roman architecture. And a lot of that has to do with the fact that we're no longer using complicated vaulting, but we're using a much simpler technology, which is wooden trusses. You get a flat ceiling because of the wooden trusses. You get a kind of thinness because the wooden trusses aren't all that heavy. And a lot of space for circulation, a lot of space for the pilgrims to assemble and for liturgical processions to take place. In fact, if you look at old St. Peter's, you'll notice that there's a courtyard on the outside and the facade of old St. Peter's doesn't even look like a facade. This thing that meets the street, this looks really like an assemblage of buildings that you might find in any city. It doesn't announce itself in a very declaratory manner that this is a church, this is a place to worship, but rather it's sort of self-effacing. So the impulse to be covert and hidden that had to do with the earliest days of the Christian church continues in the earliest of the Basilicas. Notice in plan, we have a number of martyria over here. So there is a varying of these holy figures near the church of St. Peter's. St. Peter himself is buried at the crossing of the transept and the nave. I think it's also worth saying that not only is the church a cross and this kind of elongated cross plan is called a Latin cross, but the plan of the church is also anthropomorphic. You see this little bulging out over here and that's called an apse. Let me see if I write the word there. Yes, apse. This little bulging out is called an apse. Kind of looks like a head, arms, legs. So the church becomes a body and the church becomes a symbol. But the church also becomes an assembly hall and in its liturgy, the celebration of the rights are processional. So this notion of a longitudinal church is very important so that the various people participating in the celebration of the mass can come through with incense and chanting and all the stuff that you need. There were four major basilicas established in the early Christian days. We talked about St. John Lateran and that one is more or less over here. We talked about St. Peter's and that one is more or less over here. And the two others were St. Mary Majore or Santa Maria Majore, St. Mary Major and St. Paul's outside the walls. This thing in the middle is the coliseum. This layout of the early Christian basilicas, these four earliest churches of Christendom to be pretty fascinating, particularly given the whole notion of Roma quadrata. The idea that the foundation diagram of Rome is this marking of the four square, the marking of the cardo and the deculmanus and the making of a perimeter. Here's the coliseum. If you wanted to find something that really symbolized Rome or really symbolized the persecution of the Christians where Christians were routinely put into difficulties in the coliseum being persecuted and whatnot, the idea that this becomes the center of a new Roma quadrata, a new crossing of axes is pretty interesting. So if you take an axis from St. Mary Majore to St. Paul and an axis from St. Peter's to St. John Lateran, they cross and the place that they cross is the coliseum. So just as taking the Latin cross and making the cross a symbol of Christianity, so too is the entire city of Rome reclaimed for Christianity by taking the cross and no longer using it to organize the cardo and the deculmanus, but to reorganize Rome as a new Roma quadrata of Christian churches. So if you look at the four basilicas, they look nothing like they were supposed to look like back in the old days. We have facades that are for the most part Baroque in the case of St. Peter's and even 18th century in the case of the later churches, St. Maria Majore, for example. And here in St. Paul's outside the walls, there's a 19th century façade that's trying to look like an early Christian. That's probably more accurately what the church would have looked like. It would have been a flat façade, maybe encrusted with ornament and relating to a courtyard, as indicated here. There are a couple of other important early Christian churches not in Rome proper. And one is Church of the Nativity. The Church of the Nativity takes the place that they believe the manger, this little grotto where Christ was born and commemorates the space with the church. So you have the actual historical place of Christ's birth or at least as it was believed, celebrated by this centralized piece that pulls out of a basilica-like element. So already here in the Church of the Nativity in Jerusalem, you begin to get a kind of hybrid typology of the martyrium or the centralized building and the longitudinal building. Kind of looks like the apps that we have at the end of the usual early Christian basilicas, but it's an apps that transforms and becomes more pavilionized. We have the courtyard here and all the elements that we need to establish the type. And you begin to get variations on the theme of the Church of the Nativity that happened. We mentioned that there's a split between the Eastern Empire and the Western Empire. And this little map over here begins to show you the extent of those different domains. The blue states are the Western Empire and the red states are the Eastern Empire. The Eastern Empire, as you see, is much, much more strongly consolidated into a governable whole. And the Western Empire is really beginning to fragment into bits of Italy, bits of France, bits of North Africa. And it's quite different than the extent of the Roman Empire during its heyday. And this is more or less the time of Hadrian where all of this, the entire territory, ringing the Mediterranean all the way up to England, all the way up through Germany was all intact and part of the Roman Empire. So this is an image really from the time of Constantine 330 AD on the right. And I'm showing you the Apollo Belvedere Hellenistic thing on the left about 330 BC as a point of comparison. What has happened to the attitude toward the human figure between the heyday of the Romans and the Greeks and the late period? And I think there's a significant difference. If you look at the Apollo Belvedere, there seems to be some idea that the human body, this thing of beauty, exemplifying kind of ideal perfection, exemplifying the divine and its perfect proportion, marvelous contrapasto, activating the space around it and really becoming something that is not simply static and to be viewed from one position, but that becomes a kind of player in the space of the world. The statue of Constantine's sons is a really different kind of thing. If I had to say what it looks the most like of all the statuary we've seen so far, I would probably say Egyptian, which is shocking, right? We're going way back. And the reason I'm saying that is that these don't seem to be figures operating in the space of the world. They seem to be like totems. They represent people in a kind of iconographic sense, but they don't represent people in a physical sense. They're embedded in the block that they're sculpted from rather than liberated as figures in the round. And the faces are impassive. The faces have no portraiture or personal expression, nor are they particularly idealized. So the attitude toward the human body as something that carries meaning for society has shifted. If in classical thought there is some kind of humanistic notion well represented by the Vitruvian man that the body is a microcosmic reflection of divine order and therefore something to be celebrated. Even by the time we get to late Rome to the time of Constantine, the body has become something kind of reviled or maybe not reviled, but at least it no longer carries meaning the way it had before. Meaning becomes more attached to spiritual things. So here is a description of the spread of the Byzantine Empire in the sixth century. And you can see the orangish stuff showing you the spread of Byzantium, but really the strongest centers of the Byzantine Empire still cluster around Constantinople. This is Rome again, of course. This is Rome with its seven hills, with its low ground in the middle that essentially became the Forum. And it's interesting to look at Constantinople, which Constantine deliberately established as a new Rome and tried to name Rome and to see ways in which it reprises some of these similar themes. There are ideas of Thora over here. There are spectacular fortification walls. And so a lot of the symbols of ancient Rome get rebuilt in Constantinople. The architecture shifts under the Eastern Empire from the kind of grand engineering projects we looked at when we looked at the Western Empire, the consolidated Roman Empire, to things that are more modest in scale and things that deal with the materiality of the wall in quite a different way. The thing we're looking at right here is a tiny little mausoleum for Gala Plakidia in the town of Ravenna, which is on the Adriatic coast, about 60 kilometers south of Venice. You look at the plan. How would you guys describe this plan? Anybody have a nice word for this plan? Yes? It's simple. Good. Can anybody be a little bit more specific than simple? Cruciform. Good. Do you want to take a stab at what kind of cross it is? It's not quite a Latin cross. It's not long enough to be a Latin cross. In fact, we would probably call this a Greek cross because it's, and it's not quite a Greek cross, but if you have a centralized plan that's cruciform, that's a Greek cross. And the plan type of Greek cross is something you see a lot in Eastern Empire architecture, which we call Byzantine architecture. So yeah, yeah, this is a little bit extra long, but pretend it's not extra long. It really seems to be centric. It seems to be organized around its center, but it also has this cruciform description. But look at the wall. The wall is really encrusted in mosaics, and mosaics are these tiny little tiles, these pieces of glazed stone. And one effect of the tile is to really deny the plasticity of the wall entirely. The ability of the wall to cast shadows or to be carved or to be manipulated that we were admiring in Roman architecture has been replaced by the wall as a surface to the black elements. And to me, this seems not so different from the image-making that we noticed when we looked at the statue of Constantine's sons, which is to say it's not about plasticity, it's not about volume and space, but simply about a kind of iconography or making of symbols or registering of meaning. Really, the Byzantine age comes into its glory under Justinian, from AD 482 to about 566. Under Justinian, a number of important churches get built, and a lot of them get built in Ravenna on the Adriatic, where we saw the Gala Plakidia monument. In terms of planned typology, a lot of the late Roman stuff that we looked at when we looked at Minerva Medica or St. Garion are going on here, too. That is to say, there's some notion of the centralized plan, but the centralized plan that's become complex and elaborated in intricate ways. So here, when we look at San Vitale in Ravenna, we see, at a big scale, elaboration of this notion of the silicon section within a round form, but also the kind of transformation of that with an extension that recognizes the axial nature of Christian liturgy. Here's a cutaway of the San Vitale church. You see a big dome up here, and it has the same kind of wall treatment that we saw in Gala Plakidia. Mosaics cover all the walls, and the Mosaic program here is showing you the courts of Justinian. The lad is a pancake, strangely proportioned, really kind of more hieratic than human in their description. So, when we come back next time, we'll talk about probably the greatest monument of Byzantine architecture, and that's the church of Ayasotia in Constantinople.