The shape that should be impossible.
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- Опубліковано 21 вер 2024
- Get your Sydler-shape limited-edition dust jacket now! mathsgear.co.u...
Here is the bonus video with all the details: • Making a dust jacket f...
See some of you at the Copenhagen Developers Festival, 28 August 2024
cphdevfest.com/
Thanks so much to Paul Catherall who made these beautiful prints. www.paulcather...
If you order Love Triangle on Waterstones you will not get the dust jacket but you can use the use the discount code 1+2+3+4+5+6=21 to get 21 percent off the already discounted price. www.waterstone...
All UK options: www.penguin.co...
All USA options: bit.ly/3wCTesR
Matthias Goerner's page about Sydler's shape: www.unhyperbol...
Or see the 3D model directly here: sketchfab.com/...
Henry Segerman's video: • The pi/4 polyhedron
Robin's 15° shape: sketchfab.com/...
CONJECTURE DISCLAIMER:
We believe that the "there must exist a single-angle polyhedron for any angle θ with an algebraic sine" conjecture is true but don't have a nice proof. There seem to be all the required parts for the proof in papers by Sydler and Jessen, but it's not fully assembled.
Huge thanks to my Patreon supporters. They keep me orthogonal. / standupmaths
CORRECTIONS
- None yet, let me know if you spot anything!
Filming and editing by Alex Genn-Bash
Written and performed by Matt Parker
Produced by Nicole Jacobus
Music by Howard Carter
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinran...
UK book: mathsgear.co.u... - Розваги
Pro tip: Turn this shape into a fashionable hat to ensure self-driving cars always see you.
Awesome Radar signature on that thing.
Found the radar tech
*always see you from most directions.
A little bit of risk makes life more exciting. 😉
Yeah, and then they strap a deep learning model to it and it thinks youre a duck or something
one of the most clever comments i've seen
I don't know how radar signature works but if it requires right angles and/or flat surfaces, couldn't you just use a cube? Or any simpler shapes with all right angles?
sticking the 45 degree angles of two sydler-shapes together creates the most remarkable looking shape without any particularly remarkable properties
I'm sure there is some remarkable property that we are yet to uncover. It's like the least notable number paradox
.
its remarkable property is that it can be split into two shapes that have exactly one non-right angle!
@@vari1535 Wait, can we use that idea to take an all-right-angles shape and come up with a slicing method to generate other angles? (As I'm thinking about it, I don't think so, but maybe?)
@@MAlanThomasII The other sides of the cut have to be exactly flat relative to the cut, so that's the puzzle to figure out.
Speaking as someone who was also invented in 1965 I can only say I'm very proud!
Welcome to the club 😀
If that shape is named the Parker Polyhedron, would you call yourself a Parker Person?
Do you also consist of mostly right angles by any chance?
I was 1year when the paper came out. I read it, thinking "but the shape is too complicated!" so I started screaming and my mother soothed me (also changed my diapers)
Great year. Lots of cool stuff happened then
Next you'll want to build funny shapes on top of your actual house. A Sydler on the Roof, as it were.
James Stewart moment
If he were a rich man...
Best laugh in weeks!! Thanks. How does Matt come up with these things?:
All day long, he biddy biddy bums.
Ya ba dibba dibba dibba dibba dibba dibba dum.
Well done. Rare that a pun is so à propos of nothing that it goes beyond groan-worthy into impressive that you even concocted it
A Topol-ograhy joke!
The last step of the original has Rest-of-the-f*cking-owl energy
How so? It's just a bunch of right angle cuts to navigate and connect the two offending angles. It looks complex because it's a weird shape but there's nothing to it. You can could make any combination of right angle cuts that connect the two angles.
Ha! That's what I was thinking when I saw all of the lines on the face. "They're about to draw the rest of the f*cking owl, aren't they?"
Had no idea what that meant. Coming back an hour later to say thank you for the laughs!
"This is currently my favorite shape"
The Klein Bottle can hear you, Matt.
115 likes and no replies? Let me fix that.
The Klein Bottle is a _bush league_ shape! Anyone can be interesting if you add an extra dimension or two to get around physical constraints!
@@NoobixCubeOr if you self intersect. An additional dimension is only necessary if you forbid self intersection
4D Sydler shape that is also like a klein bottle?
@@NoobixCube Hey but - this shape added an extra dimension to get around the constraints of 2d!
I'm finishing off my PhD and have about 15 references to papers that were all done in french in the 80s... I'm now very good at reading and translating specific french optimal stopping theory...
Learning French is actually applied math
In my field of metrology (science of measurement), a lot of the documentation and historical decisions are in French. It's always fun using a 120 year old white paper as a reference.
I had the opposite experience -- I accidentally learned a lot of math while practicing French because math instruction was easy to read and understand when spoken lol
1:22 Well, personally I see a lot of 180° angles in that shape, you just need to squint right.
Those are just two 90 degree angles in a row ;)
@@unvergebeneidlook closer. It's six 90° angles
But... If you are squinting _right_ then ... _confused 90° noises_
as someone who is often reliant on text to speech to communicate it's really nice to see an interview with someone who (while only circumstantial) also is reliant on these technologies!
It was really cool to see Robin participate in the video using text to speech. I have some hand injuries I am recovering from still (tendonitis). I am still adjusting to integrating speech to text into my wife and workflow. I wrote this comment using speech to text while my hands were hurting today. Accessibility and representation for the win! Good on Robin for looking out for their health too.
Thanks!
I wasn't really looking out for my health, it's just that I literally can't physically talk. (Viral laryngitis.) it's been three weeks now, and the novelty is wearing off, though it is rather peaceful.
Was "into my *wife* and workflow" a "typo" by your speech to text...?
@@gerryiles3925 it was yes
@@gerryiles3925 Presumably it's supposed to be "life", which sounds pretty similar to "wife"
@RobinHouston Oh wow, that sucks. As someone who processes things by talking about them out loud, losing my voice would a nightmare.
Random aside: people always say they hear their voice in their head when they think, but I actually imagine myself speaking when I think. And when I'm thinking really hard I'll start unintentionally moving my mouth and tongue as if I'm saying my thoughts. That's not nearly as weird as the people who don't have an internal monologue at all, though. I wonder if there are people who think by imagining themselves writing/typing or by visualizing their words as text. Or maybe someone who is deaf and/or mute thinks through sign language, so when their alone they will slightly move their arms and hands as if they are signing. That's a risky one, though, because what if they accidentally start signing their thoughts in public without realizing it 😅
Anyways, hope you get better soon and can get back to talking!
Sydler really went and made a Parker Polyhedron
Considering the parker square that makes me think a parker polyhedron would be a one dimensional sphere, but only viewable from above....
Or a Parkerhedron…
how about a parker cube?
@@jakobdiehn6596 a Parker cube is a Parker square, just one dimension higher. Incidentally, a Parker pyramid is a Parker square, one dimension higher, pointy end first.
Couldve just been calling that one non-right angle a wrong angle. Or if you like, the non square angles could also go by "Parker Square Angles"
I was thinking left angle myself, but either works...
The Parker Perpendicular
Ooft
how about "sinister angle"? plays well with the brutalist nature of the shapes
@chriscraig6410 as long as you don't go along the sinister angle on a late summer night
You've betrayed the rhombic dodecahedron! Your wife must be nervous!
I bet it will turn out to have been a one night stand, this.
I was in shock when he called this thing his "favorite shape". This "Sydler shape" doesn't even tesselate. It doesn't even make nice dice. The rhombic dodecahedron deserves better.
I'd like to see a top 10 list of Matt's favorite shapes.
His favorite polyhedron changes as often as his favorite number
@@LeoStaley He's polyhedron-polyamorous.
You know those puzzles that keep you stumped for ages, and then finally the answer is revealed to you and you’re like, “that’s so simple. How did I not see it?” This was not one of those.
Math teacher: calculate the area of the shape.
The shape:
A shape made from alright angles and one extraordinary angle...
"What are the newest shapes?" makes sense now. 💀
I had the same first thought 😂 Man was ahead of his time
Essentially it seems like the key is to remove the second non-right dihedral angle by reducing that edge to a vertex, like the one in the orange shape where 5 faces meet. I have a feeling that if many more people started looking into this we will get a huge assortment of shapes!
So pleased to see the parker square hanging out there in the background, casually representing giving stuff a go!
6:27 "As far as we know, the first person to ever make this shape was someone named m-"
I expected you to say "Matt Parker" 🤣
But it works, Parker solutions dont do that
How do we know that Sydler isn't a woman? "They." 🤦♂
@@FLPhotoCatcher rzeqdw didn't even use any gendered pronouns, if you are going to be angry, at least actually make sense.
@@FLPhotoCatcher The only person I was referring to is our esteemed host, and he is clearly a dude. Please take your trolling somewhere else, UA-cam made comment trolling illegal like ten years ago. Get with the times!
@@FLPhotoCatcher Jean-Pierre Sydler
As someone who regularly has to use TTS, it was really cool to see it used without any stigma in this
@12:45, not only is it possible to make that shape out of cardboard but I bet Dr. Katie Steckles can do it by folding a sheet of paper and making a single cut
Next step is to get the Sydler-shape used in the UK on road signs to show soccer stadiums!
Get one made in leather, and go for a kick-about. Maybe at Spurs...
i misunderstood the qualifications to be that the _faces_ of the shape had to only have one non-90° angle and i was so confused by the two shapes that clearly had faces with several angles that were not 90°
Me too, momentarily. But the faces are 2-D polygons, and we got shown that the 2-D analog is impossible.
Matt is such a good interviewer. I think this every time he has a guest. Would love to see Robin back when he's fully healthy.
Big agree! (I think Robin’s pronouns may be they/them like “they have a beard”)
Matt’s Copenhagen Developers Festival talk: “Terrible Python and Excel Abuse: The Future of Programming”
I've been binge watching Star Trek TOS recently, and that literally looks like it should be a central plot device in an episode.
There was a Star Trek TNG episode featuring a geometric paradox created specifically to destroy the Borg collective
Plot twist: Robin is actually a ventriloquist.
They know how to throw their voice out.
Matt tried to pull a Steve mould by explaining the problem in 2D
Love seeing augmented/alternative communication used for interviews and things!
I hate it
@@planetfixerInteresting.Why?
@@martinmckee5333 It is potentially distracting in high-level concepts (no offense to Robin). The smartest people are not always the most effective communicators, there is a sweet spot usually. Think of good teachers in your life, for example.
@@martinmckee5333 super performative - this person (who apparently uses they/them, which isn't bad on its own but in this case they're clearly a fan of being treated specially so I do question why they use the pronouns) "lost their voice" and instead of just doing a written interview has to draw massive attention to their affliction with a Steven hawking robot.
It just reeks of Munchausen syndrome attention seeking in this case. No issue with it in cases where it benefits people who need it, but Im uniquely sensitive and dispassionate towards this specific type of individual who's clearly just leveraging it to get the residual positive attention from simply being adjacent to a tool that has genuine beneficial applications
@@planetfixer I'm not sure what makes you feel that way (that was not the impression I got). But I can certainly understand being annoyed by it given that impression.
How on earth did he design this shape without having a physical model?! Some people's brains amaze me. I mean, when I was a kid and they were testing me to see if I should be put in gifted classes, one of the areas I scored extremely high in was '2D/3D spatial awareness' but I can barely begin to wrap my head around doing this purely from a theoretical point of view. Unless he didn't even try to keep it in his head and instead did it all mathematically somehow.
There's projection (i.e. drawing the 3d shape from different angles) and there's also being able to visualize.
Since I have issues with writing, I became excellent at holding complex shapes in my brain, so I could do something like that without paper and there's probably other people who can, if you break it down it can become simple.
I wouldn't be surprised if it's that, that it was purely mathematical!
I'm sure that given enough time, anyone could come up with this. It just comes down to thinking it's possible and worth it enough to spend enough time doing this.
I think spending time working with 3D modelling or CAD tools is a very good way to develop this sort of skill.
OMG. The framed Parker Square was there all the time behind Matt in his Parker Orthogon video and I didn't even realize it was there until I saw the Numberphile video where Brady gave this present to Matt!!!
Really interesting hearing Matt talk about the difficulties of building the original sydler solid out of cardboard, because the first thing I did after watching Henry Segerman's video was building a paper model of it (I don't have a 3d printer and Blender's paper model export is awesome). Yes it was quite finicky to put together, especially the funky recess and gluing the last edge. The one I built is the mirror image of Matt's 3d print
Reminded me of a riddle
Start at your base camp looking for bears
Go south 1Km, no bear
Turn 90deg go east 1 Km , spot a bear
Turn 90deg, go north 1Km to return to base
What colour is the bear
On a spherical surface you can have a triangle with angles adding to more than 180deg
Classic non-Euclidean riddle!
I'm going to say the bear is black, because a ~1km radius sphere wouldn't be very good at holding atmosphere, so the bear is probably cooked to a crisp. Alternatively, an artificial habitat that had bears may want less aggressive bears, so black again.
@@TlalocTemporal very clever. Of course if the poster didn’t specify turning 90 degrees and simply used cardinal directions then the radius wouldn’t have been implied and the riddle would have worked as intended.
@@diggoran -- The turning angles don't actually constrain the radius, and now that I think of it the distances don't really either. If you had a perfect compas the second leg would curve to follow due East. I made the assumption that each leg was a straight line, which can only happen if you're traveling around the equator.
As the OP has described it, you would return to base from a direction ~57° East of the direction you left from, assuming the sphere you're travelling on is much bigger than 1km. That reyurn angle becomes 90° when you travel exactly 1 quarter of the circumference, and could be anything when traveling about half the circumference. However OP said nothing about this return angle, and North/South & East/West are always 90° perpendicular, so we know nothing about anything except the base is at the northernmost point.
Thus I'll say the bear is red, as it has eaten me. :P
@@TlalocTemporal You’re right. At first I thought the turning angles do constrain the radius of the planet because as the planet approaches infinite size, the path approaches an equilateral triangle on a flat plane, so the turning angles (e.g. from due south to due east) would be 60deg. However the formation of a triangle was the flaw in my logic. On an infinitely large planet, the walking path would actually be a sector, with two straight edges but still one curved edge, as in order to always walk east, the walker would have to continuously turn slightly left and stay equidistant with the North Pole. All that shrinking the planet does is distribute the curvature from only the west-east path to all three paths.
me: "its in French how could you read this?"
matt: "MATH IS MATH"
To be fair, the shared roman alphabet characters helped a lot too
What mainly helped is that most of the english terms in the title had their etymology from french
@@arnspyarchi6040 About 40% of the language is imported from Latin or French, which is why English speakers can probably understand more French than German despite English being a Germanic language
It's more correct to say the words have shared etymology, rather than directly taking their etymology from French.
The text to speech was so soothing.
Real Ziggy vibe with the line breaks.
10:20 THE FINAL SHAPE AHHHH- Somebody call The Witness!
Brilliant timing for the video release tbh! I did a double-take when he said that
Maybe Matt is one of many Disciples of the Witness?
Loving the parker square in the background
we now know it's the original too!
Brutalist geometry
Building blocks misunderstood, mistreated and re-visited
lmao
What amazes me more is the number of years between the discoveries of Csaszar and Szilassi polyhedrons.
This is a really nice demonstration of how technology can improve research: having access to 3D modeling software and computers in general is crucial for these kinds of discoveries. And hopefully some of these discoveries will eventually lead to even better technologies in the future too!
Man I remember reading about Sydler over a decade ago and thinking, "Oh that's neat." Forgot entirely about it until today. And you know what? Still neat!
Can't wait for the transparent LED version with Adam Savage
As a machinist this does not surprise me at all. I once spent like an hour trying to figure out why the heck a thing I had just made was not right even though all the angles were square.
Now someone should come up with a shape that has *only* 90° angles. None of this imperfect stuff with an angle that's not non-right!
📦
that would really be something! Mathematicians just don’t come like they used to 😤
"the final shape"
Well that is coincidentally currently a relevant phrase
Why's that? Not just for out-of-the-loop me, but for future viewers too. :Þ
@@NashRespect"The Final Shape" is the name of the Destiny 2 expansion that released yesterday
This one has way fewer screaming faces and amputated hands, but it’s still cool.
4:36 Wow, I guess maths really is the universal language
sidler shape… right…
first thing i thought is: NO! there has to be a much much simpler shape. this is completely random and complicated.
I wonder if the new shape the minimum number of surfaces? Robin has hugely optimized the shape and it's very impressive. I might print both shapes myself and put them on my desk as art.
If you turn the shape upside down so the 45 degree angle is on the bottom, and you attach a handle to the opposite side, you have the world's most mathy bottle opener - I swear it's the perfect shape for it if you make it the right size. Robin should make that and sell it, I bet it would make a killing!
Doesn’t Robin’s shape have a triangle as one of its sides? And doesn’t a triangle necessitate two non-90 degree angles? I’m confused. Actually, there are a number of sides that appear to be triangles.
Not the edges meet in 90° angles, but the faces.
@@Doeniz1 ohh okay, thank you!
This shape has never read “Blame!” Also, Matt, why are you such a prodigy? You and your friends are crazy, and crazy good.
Robin must be an awesome Dungeon Master!
For those who are confused, it is the angle between the faces, aka the edge, not the angle of a face.
it's a trivial problem just put a 180 degree angle in a pentagon and now it looks like a 5 sided square
?
You're mostly correct about the existence of a 2D object having one and only one angle that's not perpendicular. You can't make one in a Euclidean planar geometry. In a hyperbolic plane instead of rectangles you can get either Saccheri Quadrilaterals or Lambert Quadrilaterals. A Lambert Quadrilateral in a hyperbolic plane will have one and only one angle that's less then 90°. Note that if you construct a perpendicular bisector on a Saccheri Quadrilateral You'll find two Lambert Quadrilateral.
"Hold my shape" - Sydler
10:23 The Witness - ahh yes the FINAL SHAPE I've been looking for!
I have nothing useful to add here, I just came to say "Recreational mathematicians writing terrible python code."
"I don't speak French but I do speak mathematics"
Well do I have news for you! You speak (some) French, we pretty much coined all the modern mathematics vocabulary during the renaissance and industrial revolution :D
isn't there two 45 degree angles in the front bottom of that orange box? It's the corners though, so that somehow doesn't count?
None of the face angles count, only the face-to-face ones. If you constrained yourself to only right angles on the faces themselves, it would truly be impossible just like the 2D version.
In 2D shapes, an angle is between two 1D lines.
In 3D bodies, an angle is between two 2D faces that are adjacent, like if you folded them. You don't look at the angles between edges, as that's going a dimension lower.
I think, thanks for listening :)
The angle between two adjacent edges is not constrained, we only care about the angle created between two faces. This admittedly took a while to "click" for me, but this is the simplest way I can think to explain how it suddenly made sense.
You could probably do it in 4d then? XD
Take a cube, chop one corner off, done. Maybe i misunderstood how the all right angles,but one is not, is supposed to work
I am an arts and humanities student but I can’t help loving some of your videos, this one was especially interesting! Shapes are amazing
I have little idea about maths though
Me, adding a 180º angle to a square
it makes sense you can do it in 3d because you are allowed to have vertices that are not 90 degrees. this means you can twist the surface on a vertex to get the required dimensions to get back around to the start with only 90's.
2:49 Technically it's not impossible in 2D. You can draw a triangle with two 90° angles and a different one on a sphere. It's only impossible in 2D euclidian space.
On a sphere you can even make a triangle with two 90 angles and one 180 angle. So a two sided triangle 😊
Just went to remodel that in SOLIDWORKS and discovered that the regular looking trianges in the cavity are in fact EQUILATERAL! very elegant.
Check out Jessen's orthogonal icosahedron, if you don't know it. It has eight equilateral-triangular faces, and it's related to my shapes in an interesting way
Interesting question, is there a convex shape with this property? I suspect no.
You are correct! There are no convex polyhedra like this. Also, a convex polyhedron *all* of whose dihedrals are right angles has to be a box.
The paper that came with Robin's Gift Exchange gift was one of my favorite ones to read. I have their shape proudly displayed on my shelf at home.
That's so nice to hear! Thanks.
missed oppertunity by the math community to call it sidlers solid.
The illustrator is genius. Paul Catherall? Holy heck. You did good!
I missed the original framed parker square on the back
Robin seems awesome. As others have said, Matt, you're a great interviewer. It would be lovely to see you sit down and talk with Robin someday hopefully when his voice returns.
Edit: preordered the book long ago. Can't wait.
i've only seen robin in this video and thus might be wrong, but matt seems to exclusively use they/them for them in this footage, so i suspect they're not a him
emphasis on Euclidian space. The moment you try this in non-Euclidian space a shape with these properties becomes quite easy, even in 2 dimensions.
Yes, impossible in 2D euclidean space. But I think I've met some orbifolds that tell me that not all non-euclidean spaces will allow a solution. I would guess any curved, locally euclidean space always has solutions, but they're not the only spaces that do.
it's a great example of how we resist change in science, i mean clearly it's not impossible, the hypothesis was simply wrong to begin with :)
Some of these shapes remind me of Elite ships (original game wire frames).
I figured out a way of doing it in two dimensions! So you just need five 90° angles and one 270° angle. Because only 90° is a right angle. 270° doesn't count.
In two dimensions, I jumped right to a fractaled infinitely shrinking series of 90 degree angles to close the gap between the true non-90 degree angle and the "false" non-90 degree angle made up of shrinking angles. Feels like you could work that... just not (really) in real physical space. :/ hmm
Looks like bot stole your comment. :/ In other news, clever solution! It reminds me of "wild knots" from knot theory, infinite series of shrinking knots (though I don't fully remember how they're defined).
I was very surprised the first half of the video because intuitively one can think that there had to exist a simpler, symmetrical volume that works. Happy to see it at the end. 😄
4:50 SZA CONGRUENCE??!!
I've only rarely gotten that excited about a book. It would be good to grow a library again.
I find it hard to believe that a new shape was made up mathematically in the 60s and NO ONE put one together EVEN DIGITALLY until 2021?? Like how???
Not even out of curiosity???
It's not that you speak math, it's that a bunch of English comes from French and Latin (which is really old French).
That's a pretty alright shape!
Everything is right angles, you can't have right angles, just two, but all is right angles. This is beyond the most complicated video you have ever produced!
Let me try and convince my mother to buy this book as a birthday present
Mission successfull- pre order just completed
20:30
Very SUPERHOT-looking!
To clarify, the angles being measured are all in 3d, because there's a lot of non-90 degree 2d angles in those shapes.
You could say all the angles (but one) have a minimum or maximum measurable angle of 90°.
What exactly is a "2D angle" in a 3D shape?
What exactly is a "2D angle" in a 3D shape?
@@karlhendrikse A 2d angle on a 3d shape is formed by three vertices in one single surface. A 3d angle is composed by two adjacent surfaces sharing one edge, like a butterfly. The body of the butterfly is the edge, and the wings are the surfaces.
@@karlhendrikse -- A 2D angle is the angle between two lines. A 3D angle is the angle between two planes. Every 3D angle has a 2D angle of the same size along the same acis, but you can skew the 2D angle around a different axis to get a different 2D angle that's still along the 3D angle.
Imagine holding a square up flat in a 270° corner (like the corner of a room). It's 90° will fit perfectly in place. If you twist the square so it's no longer level, the square now has room to roll back and forth between the walls. If you keep rotating the square until it's completely vertical, you can fit the edge of the square (a 180° angle) right up along the corner (a 90° angle along a different axis).
Thus you can fit a 2D angle that's closer to 180° along any 3D angle if you misalign them. As you twist the alignment, the 2D slice matches the 3D angle less and the line along the crease more, which is a 180° angle.
I hope that helps, and I hope angle still sounds like a word.
_pauses at __0:34__ to study it..._ It has at least *2* non-90° angles. Besides the advertised acute angle on the outside, there's an obtuse angle on the concave side of the bit in the middle. Aaand on a 2nd look, I'm annoyed because one of the walls from this angle goes up to another non-90° angle, cancelling out the first one. The angles could have been correct, that wall could have been thicker, non-tapering, and then I wouldn't have needed to write all this out. humpf!
Where can we find the nets of Robins shape, to make it ourself out of cardboard?
Two challenges: 1: Do it in 4 Dimensions. 2: Explain it in APOUD :P Really nice
Me halfway through: I can't shake the notion that there should be a more elegant shape.
correction: matt, your 3d printed shape is wrong. the model has a little sliver in a corner somewhere which is so thin that the slicer that turns the 3d model into g-code ignores it and it does not get printed. without this sliver, there is an additional non-90-degrees edge, but it's easy to miss because it's only off by a few degrees and very small. I found out because I decided to print it myself and check all the edges because something just didn't seem quite right :D if you look at the thumbnail it's near the bottom left corner of the model. that said, I love the video, this is all very cool.
just noticed you're the guy who wrote "Humble Pi" and "Things to make and do in the fourth dimension"
Cool. Was going to ask about angles other than 45, but you answered that by the end of the video. I'll see if I can find more!
Another puzzle for anyone interested: can you make an isohedron with only rectangular faces, aside from the special case of a cube? Isohedron means face-transitive, ie all faces are the same, and fit symmetrically the same way into the whole. Faces are allowed to pass through each other too. Turns out there are 4 ways to do it.
12:39 "Imagine trying to make this out of cardboard" - I smell a Patreon exclusive video...
Could this "niche" area of mathematics find appliance in material science like with regards to crystal lattices or something like that? I'm excited on what new possibilities are awaiting us thanks to this field!
0:19 What's that over Matt's left shoulder?!
I’m back to this video to appreciate the framed brown paper with the Parker Square!
I mean... it looks like there are multiple non 90 degree angles on there. Going over the whole shape in detail would be appreciated!
Yeah, I'm also confused. I see a lot of inverted angles which are clearly not 90. I wonder if it's because of how we see it on a video or are there mistakes which are just ignored
It's the dihedral angles that are all 90°. I imagine you're looking at the face angles (which are mostly not 90°)
The 90⁰ angles being referred to are the 3D angles between adjacent faces, not adjacent edges. There are of course lots of triangles, all of which have non-right 2D angles, and if you constrained the problem to 2D angles as mentioned at the beginning, it is actually impossible.
But if you look at each fold line, the angle from one face to the next is always 90⁰ (with the one exception). It's definitely more tricky to see through a camera than in-person, because you're looking at a flat projection of the shape.
@@miorioffin 2d we are interested in angles between lines. Here (3d) we are interested in the angle between faces. There are many angles between lines that arent 90 here but we dont care about those as its 3d
Ah, so it's just defined differently. Thanks for explaining!
these prints are amazing concrete abstract art ! Loved seeing them, gave me ideas of paintings to do