The shape that should be impossible.

now we completely eliminate the 109.5° edge by cutting out the octahedron that we made earlier on the right, so you remove the bit that you designed a second ago on the right and you are left with that and look what is left this fun little Decadron where all dihedrals are to the right. angles, except this one which is 135°, which is starting to look a little like this one, but like the missing one, yes, such a move we need a final move to get the extra edge to be 45°, what 135 is there? There are a few different things you could do, but my favorite is to notice that this

shape

is like a box from one side, so what if we complete that box and look at the complementary part and that's it, so you designed the void first and that works, but it's 135 and when you get the complement of the rest of the box you get a single 45, that's correct.

What I like about that is a lot of your stuff above because I mentioned super permutation a moment ago, it's writing code and waiting a long time for the computer to find something, whereas this was just sitting and thinking about it, a lot of questions and sit down, that's absolutely fantastic and do you have the BR version of this that you carry around for fun? You

should

ask. I knew it. ask because Robin always has this brass one, look, look at that, so it's a solid brass version of this, um, a little smaller, a little more expensive than cardboard, and what's good about it.

Remember we were both at the Gathering for Gardener conference? and I took some photos of your brass thing in the... we sat next to each other for one of the sessions... uh, my love triangle book, this is, this is the first copy, it just arrived in the post um gosh a couple of days ago I talk about your

shape

congratulations thank you very much um I talk about your shape there and those are the photos that I took at the table at the meeting for Garder of your shape and I managed to find out if I tagged the 45 as the simple 45 degree angle at the top, I mean, one page before that is the 3D print that you gave me of the Sidler shape, so that's the 3D print that you gave me of the Sidler shape, I took some pictures of that and then, um, nice pictures, yeah.

Thanks, most of them are made by professional photographer. I did this one at a conference, so you know, it's a little confusing. Sorry about the whole voice thing. There is no problem. Thank you very much for joining us. That's how Robin was able to go from this to this by thinking it through. The reason I included Robin's and Sidler's forms in my book is that I wanted to dispel the notion that geometry is something that happened millennia ago, like all the Greeks did. of this and now we just read about what they did, no. I have a whole chapter on how we're still discovering, inventing shapes, geometry is still happening and there's more to do with this if you want to get into the details in a moment, but I was so excited about this shape that when I started pre-selling love triangles, if you buy it on Amazon or with the Waterstones discount code in the description you'll get this book, but I was like, ah, the people who buy it.

Well obviously they get a first edition and I sign it that's great I thought wouldn't it be good if I could add something else something that I think really represents the book because the cover is wonderful it's very it's a cover designed for a generic bookstore I think which is great, the people at Penguin did an amazing job. I thought, what if I made a really nerdy cover just for the people who pre-order the book through me? Now there's a print artist called Paul Cathol who does fantastic stuff. Print Work often based on brutalist architecture. I've been a big fan of his work and we spoke a couple of years ago.

I invited them to one of my exhibitions, so I contacted Paul and said, look, this looks a lot like brutalist architecture. Could you make one of your architectural prints but based on the Sidler form as if it were a building? I'm so excited I haven't had a chance to get them framed because they literally came in the mail a couple days ago and we came. running down that's why this is not a great print because I thought I had to film this so we finished the print and everything and here we are and here we go oh, I present to you the Sidler form as an architectural print made by Paul caol and not just in one color, let me pull out a couple of these, so Paul made them in three different color combinations, um, there's the one done in red, how cute, look at that and you can see the 45° angle, like this which is like the leaf over here, but then you look at the shadow, so Paul said they took this.

I actually had like a little red model of this on my desk before I posted it, which Paul took outside on a sunny day and looked at the Shadows being cast and you can see like in the corner there that 45°. I don't know if that was deliberate, but I love the fact that there's your 45 Dee angle, a shadow cast from there, so that's the red Sidler red Paul also made. Sidler Yellow, which is this one here, look at that and finally we have Sidler Blue and the reason why Paul did three different color combinations is because I said look, I want to do these um with the theme of triangles in different numbers of dimensions and So I said: I'm going to make a Simplex, a 4D triangle path that will be the yellow one.

I'm going to do a tetrahedron tour which is going to be the red one and I'm going to do a triangular edition, uh, dust jacket, based on the Blue Print so what will these be, so if you pre-order my book through Mass Gear, you'll receive a dust jacket based on one of these designs. I think Paul will sell a lot of these prints too. If you go to the second channel, I'll link it. I'll have all the details on Dust Jackets pre-orders and everything else below. Watch that video, but for now, as I promised a moment ago, there is something you can help us with. 45° is not the only possible angle you can combine just one with many other 90° angles, in theory any angle that has a sign of that angle, so only the proportion inside the triangle is an algebraic number, i.e. , is the root of a polynomial with good coefficients in In theory, you

should

be able to construct a form that has all right angles except one angle that is not a right angle as long as its sign is algebraic.

Robin has found a few more, so they found one with a 60° angle, not just 45, they found one. with a 15° angle, in fact, I have the 15° angle here in a 3D model that Robin made so you can see it, it's like that wedge, there's 15° and then the rest of this, everything around it and you can see similarities with its other shapes all of this is just the machinery needed to get rid of that to get rid of converting that into a 90° angle so everything else is 90 and we have that 11 15° there we don't know what other shapes well we know theoretically what can happen we don't know how to construct most of these shapes we only have a few small examples this is a very new niche area of ​​mathematics so we would like to hear from anyone who can find a shape with all right angles except one not right angle, we don't really have a format where you have to submit it, you can submit it like Sidler put it in the original document where he just explains the steps roughly how it should work.

You could make a 3D model and send it for us to see. We don't care as long as it's human processable and you explain how to build the shape. link to some   examples below so you can see them send it to us, we don't have an official website   or anything like that, if you just email it to sidla atst standupmaths tocom, that will be automatically forwarded   to uh actually, to all that I have known. mentioned in this video so we can see it and I don't know, at some point in the future we will update a website or follow up on the video, it depends on how interesting the results are.

I mean, potentially, someone could come. Create an algorithm, write some terrible Python code that can generate shapes that have this property as for everyone. We know that there is a master way in which you can, by algorithmically adjusting other bits, you can move through every angle imaginable. We have already written some. Terrible Python code that can check it if you submit it in certain 3D formats, so I'm going to link it below if you think you found a new way, email us. I'd love to see what else is out there. many new shapes waiting to be discovered.

Thank you so much for watching and thank you to everyone who has already pre-ordered the love triangle. I really appreciate the link to the other video that I'm about to film where I explain all the details about the pre-orders uh the mistakes that that made in the first edition that won't be in later editions the dust jackets the printings all of that I'll link to below and that's all thanks for watching and thanks for pre-ordering