A New Discovery about Dodecahedrons - Numberphile
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- Опубликовано: 26 сен 2024
- Featuring Jayadev Athreya on a new discovery about platonic solids, in particular dodecahedra... Extra footage: • Yellow Brick Road and ...
Yellow Brick Road Sticker and T-Shirts: teespring.com/...
More links & stuff in full description below ↓↓↓
PAPER: A Trajectory from a Vertex to Itself on the Dodecahedron: arxiv.org/abs/...
PAPER: Platonic solids and high genus covers of lattice surfaces: arxiv.org/abs/...
More great resources on the project: userhome.brookl...
More on Numberphile...
Platonic Solids: • 5 Platonic Solids - Nu...
Platonics is higher dimensions: • Perfect Shapes in High...
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Learn more about MSRI in this interview David Eisenbud: • A Proof in the Drawer ...
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoun...
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Videos by Brady Haran
Animation by Pete McPartlan
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Extra animations: ruclips.net/video/hpNPPqm1EMc/видео.html
Yellow Brick Road Sticker and T-Shirts: teespring.com/yellow-brick-numberphile
More resources on the project: userhome.brooklyn.cuny.edu/aulicino/dodecahedron/
How was your day?
G'day
Yo Numberphile what happened to James Grime
How LONG did it take to make those animations? You really did a great job!
A triangular hexacontahedron is also a regular solid.
Jayadev: "Pac-Man might be a reference from too long ago."
Brady: "Asteroids!"
To be fair, Pac-Man is still a cultural icon. Most kids probably know about it still.
@@moth.monsterYeah they do, there's a relatively new animation series about pac man.
My mind did go to the half a press video
@@korenn9381 Pixels?
@@korenn9381 there is?? Why
Out on a 1st date. "So what do you do?"
"I solved a 2000 year old problem on how to avoid your neighbors. Mathematically. My tattoo will explain it."
Of course, there goes any non-mathematician date.
Oh man if my date did that, I would be in love.
@@uganasilverhand Many women would love a guy passionated about his field and able to explain it well.
@@Ceelvain the wife's always asking me to stop talking about math, computers, and science
Didn't asteroids have a hyperspace button that transported you to a random spot?
Sounds a lot like punching through a facet, traveling inside the solid, and punching out to the surface again.
The featured mathematician is a uniquely elegant and fluid communicator. Also, the animations were tremendously illustrative and enjoyable. Loved it.
including the animated Hannah Fry jogging?
macnolds you could tell how well he understood this subject by how he could break it down into discrete parts and explain each aspect simply. Very charismatic and obviously very intelligent guy. A real treat to watch
@@alveolate *especially* the Hannah Fry jogging.
Did you squint really hard when the jogger was on screen?
... Kind of
"One of the central tensions in the Little Prince, other than the loss of childhood innocence, ..."
I love how he just glosses over that
Well... he has to stay on topic
Because that is not important
12:20 Was anyone else expecting him to turn around and lift his shirt to expose a back piece of 120 pentagons?
No
@@matthuckabey007 but we did hope a little
·...·
I wasn't, but now I feel like I should have
It's the reason I watched, had a feeling
Jayadev Athreya was my calculus teacher last year, super chill dude never thought I'd see him in a Numberphile video, small world I guess.
ayyyyy a UW fellow
Shows the Rockefeller Education System works at making people unthinking.
@@pentuplove6542 What?
@@pentuplove6542 Er, what? Would I be correct in thinking that you didn't do very well in the education system?
What os his nationality? Persian? That is a strange name.
"There's the cube - everyone knows the cube."
Just platonically though, right?
JNCressey this should have more likes
BIBLICALLY
Unless you're the Curator from _Gravitas._ We all know he's dry-humping the cubes instead of making new galleries, which is why the game is so short.
-sad cube noises-
This is hilarious
I like this man. he went through insane amount of mental gymnastics to show the world his tattoo.
Ikr
@Fremen unlike a certain "generation" of youngsters (and some oldass dudes too) who mark their bodies as if they were cattle. In the modern age, one of the first groups to mark their bodies were inmates, then whores, then the fkng nazis marked our brother jews, then these youngsters now.
@@webgpu that's a bit of a slippery slope fallacy, isn't it?
@@TheGamingLegendsOfficial Yeah, it's pretty slippery, but possibly not too far off the mark.
Well, He must be better than me, I have no tattoo to show off.
I love the way he talks about the subject. It’s easier to feel interested when he makes it seem so exciting
I love how his blackboard is so intensely clean
he does everything in his head, no blackboard needed
Or he's a conman on the grift, making up shapes as he goes and glibly explaining it to get recognition and subsequent riches to pay for more body art. The conmans clean blackboard is a gullibility mirror 🤣
"One of the central tensions in The Little Prince, other than the loss of childhood innocence" 😂😂😂
( ͡° ͜ʖ ͡°)
I don't know why, but I lost it when he said that.
@@loonycooney22 I wish you luck on finding it. :D
_its a great read..._
@@loonycooney22 Oh gosh, you lost your innocence too?
This guy probably never gets invited to play D&D with his friends. "Do you guys wanna know a cool thing about this D12?"
"Here we go again..."
i'd happily abandon D&D to talk geometry with this dude
Ismail Shtewi who wouldn't?
It is statistically proven that a majority of us d&d players are absolute nerds!!!
(I'm a nerd and I play d&d so everyone else who plays d&d must also be a nerd!)
🤣
@Cliven Longsight Quick, find a mathematician who can get super-excited about antiprisms and isohedra!
Thats is why my dnd party hates me.
When a video is called "a new discovery about dodecahedrons" you know you're in for a ride
Ahem, for a walk.
@@danielalba7651 jog
@@zethyr8833 a roll
Dodecahedra is the plural.
false.
What I really find amazing is that their simplest answer is so simple that it’s easily visualized by the human brain; it’s not abstract at all like so many of the proofs or conjectures discussed on this channel. Like you could show that example to a mathematically inclined middle Schooler and they would understand what was going on. I think that’s so wonderful!
I wish someone looked at me the way this guy looks at dodecahedrons.
OOF
That's why his chalkboard is so clean, erasing all those naughty angles and planes
“How many pieces can I cut it into? I don’t know! Let’s find out!”
2:28 Academy nominates Numberphile for best animated short horror film.
It's adorable
@@volodyadykun6490 Are you blind!? Look at it!
@@namethathasntbeentakenyetm3682 and you are mean
@@volodyadykun6490 and you are trying to protect the feelings of a 3D model. Which happens to be possessed by a demon
@@Alistair that mathematician may be meant to be Hannah Fry
Never knew I'd be so interested in platonic solids and mathematical proofs. Sick video.
Well you have a truncated cube in your icon so that’s something
@@ar_xiv Its because only us cool kids use types of cubes in our pfp xD
@@DerpMuse No need to be hyper about it.
@@DerpMuse
Oi, do you have a licence for that 4th spatial dimension?
??
It didn't surprise me at all when he said they've made educational materials about this for schools, because his explanations already sounded streamlined in a "for non-math enthusiasts" sense. Not just the words themselves, but his tone of voice implies much practice as well -- even compared to the Numberphile regulars, which is impressive!
... kind of
Deep love how you broke that down
3:01 "One of the central tensions in The Little Prince *_(other than the loss of childhood innocence)_* is that the prince has a sheep on his planet..."
This made me laugh.
“Other than the loss of childhood innocence
Nice
That was what I'd suggest the path be called: loss of innocence path
He quoted Ecclesiastes too
@@Om_1337 To show that the holy book was wrong!?
The line on the dodecahedron almost hitting the points is like the DVD menu icon almost hitting the corners of the screen.
It is the exact same problem, as bouncing is continuing towards into a mirror image
Glad I'm not the only one who spent WAY to much time watching and waiting for a corner bounce to happen. I wonder if we had the same DVD player as kids?
I remember watching that and trying to solve it.
No, you should be glad that your tv was not shaped as a pentagon, because then you could've been watching forever...
Mathematician: Platonic Solids.
The rest of us: Dice
Ian Tan MY SPECIAL D12 CLICK CLACK
Ian Tan Ah, a man of culture I see. For this you get advantage on your next skill check
EXOTIC ENGRAM
@@404dne inDEEd
the d10 is my least favorite die because it’s not a platonic solid
"So what's your job"
"I imagine small living creatures on geometric objects"
"Haha... no seriously what's your job"
"This is my job for the past 25 years"
"😐"
....."get a real job"....
one where hypothetical sheep and joggers HAVE WIDTH..... then a SIMPLE brain-fart, doesnt become a 25 year challenge!,
Let's assume a spherical cow...
A genius doesn't waste the other person's time hiding deal breakers.
Interesting! I had no idea something like this was unknown until recently. Just for fun I calculated the areas of the two pieces separated by the primary "yellow brick road". Apparently they're not the same in area, one piece is 9.66% larger than the other. The larger piece is 65-sqrt(5) parts out of 120...
how did you do it
@@arnavrawat9864 Hmm, it was a while back, but if you look at the video at 6:50, you can find the areas of the two halves of the surface by carefully measuring all lengths and angles, and finding the area of a certain right triangle whose hypotenuse is the red line. I could produce a proof in a youtube video if there was enough interest, but alas you're the first person to ask! But thank you for asking.
😀
@@mnek742 Honestly, i would watch the proof 1000x, POST IT.
Well done for working this out. With the areas of the two surfaces being in the ratio of about 1·0966:1, I wonder whether the volumes of the two pieces are in the ratio of about 1·1483:1 ?
Who would've thought that there are still things to learn about the good old d12 :D
So ddozen?
Not just for barbarians anymore. ;)
@@J3rs3yM1k3 great axe wielders unite
Ah the dirty dozens
It doesn't have to cry itself to sleep anymore!
Man, I hope he gets to do a lot more videos in the future! He’s fantastic and enthusiastic!
The animation by Pete McPartlan is amazing. You guys have really outdone yourselves this time.
Edit: Congratulations on the great discovery. A name for the simplest line could be The First Bridge of Koningsberg, I don't know
Bridges of kongisbreg is already taken
@@jonathanlevy9635 yeah I know, I was saying that the original problem that Euler solved was to avoid going through a vertex (but a graph theory vertex, if you know what I mean), and this is a similar problem but with the vertex being a "real" one.
This guy is great, he is a very clear speaker and doesn't fumble words.
I could listen to Jayadev explain anything....so brilliant but easy to understand.
I wish I had 31 ways to avoid my neighbors
Anonymous S move?
Live on a dodecahedron then
Use the back door or underground exit
Skip out the back, Jack; make a new plan, Stan...
Talk to them once and a while you will find that you will see them less.
Literally the cleanest blackboards I've seen in my life.
...those _are_ blackboards...
@Hoaxcrit yas
Important problems for mathematicians: how to go jogging witgout meeting other people :)
And thus the treadmill was invented
witgout? A Dutch / English word? Wit is out and out is out.
Haha you never know when you’re going to run into a fellow Brony on the Internet!
Suddenly, this is a relevant question for EVERYONE
Next thing you know they are going to be telling us that a coffee cup and doughnut are the same thing!
I dont fully understand what your explaining but i love your enthusiasm and excitement on the subject. Shows that your really passionate for what your doing
Always great to see new faces on the show, but this guy takes the cake! His deep but light-hearted passion was infectious, accentuated by unique mannerisms that amounted to frequent full-body chuckling. A joy to watch.
And the muse of his amusement: an outside-an-unexpected-Pandora's-box-of-a-realm, in the vicinity of the fundamentally familiar Euclidean solids! Great fun, many thanks..
I've been struggling a lot lately with a burnout in my passion for math. This video helped. Thank you
I don't know any of you, but when I watch Numberphile, I feel I am among friends.
"Here's a mathematical proof*!"
*Some assembly required.
That's most proofs, but they usually word it as "left as an exercise for the reader."
@@trevorvanderwoerd8915 To be fair, that's the best way to get the reader to truly understand it. Force them to play around with it on their own.
Assembly? I thought this was numberphile not computerphile!
I watched 19 min long video about Dodecahedron, and I want more. This guy radiates positive energy. Extra footage, here I came!
2:38
Its gonna haunt me in my nightmares
Love how clearly this problem and solution is explained. Also the questions were spot on with helping to gain more insight and understand this better. Great work!
Can we get more of this mathematician? I love his explanations.
Natalie made his idea clear he made it clear that he was the only one who had this idea. Which I appreciate it. Sometimes I'm not sure what the mathematician accomplished in this case he made that very clear as well.
@@CalvinHikes Kind of...
I can't thank him enough for his amazing proof that he submitted, what a passionate teacher. Need more of this type :)
This was my Number Theory professor in college. Hands down one of the best professors I have ever had. So cool seeing him on here!
Interestingly, I was building a wooden dodecahedron and as I was gluing and taping up the shape, I discovered this very thing. Great video.
The Dodecahedron has always been my favorite platonic solid, and I feel very justified seeing how weirdly different it is from the others even though I didn't know most of this stuff.
So weird! I recognized this guy. Sure enough, he was my math teacher at University of Illinois!!
Prove it !
Ray Bois nah i think timothy was just saying where he teaches
I think you lie!!!
@@raybois what is "name dropping"? never heard that before
my first language isn't english
@@raybois thx
learn something new every day
Round earth? NAH.
Flat earth? NAH.
Dodecahedral earth? YEAH!
That'll shut them all up
You forgot donut earth ;)
"The Dodecapath" for the simple path name?
I'm stealing this for a school assignment
Videos like this are prime numberphile content. Simple questions with interesting answers and can be clearly explained in a 15 minute video.
Platonic solids are so fascinating and powerful. This is one of my favorite Numberphile videos. Great job Jayadev, Brady, and Pete! I felt a real sense of intrigue and mystery as you slowly explained the concept and revealed the story.
Love the shape animations!
Let's call shortest path "Antisocial lazy jogger".
Antisocial travelling salesman.
@@alexanderf8451 unsuccessful salesman!
GummyGruffi
ALJ
It is however, misusing the term "antisocial", and the term he's looking for is "asocial".
Antisocial: meaning one who actively seeks out to destroy social relationships (a term often used when describing psychopaths and sociopaths)
Asocial: meaning one who shys away from being social (often a term used to descripe depression symptoms)
yyyyyyyyyyyyyeeeesss! You _have_ to use/include the jogger's dilemma, to omit it is a crime!
Thank you for letting me know that my sleep paralysis demon is an avid jogger.
When I thought of living on the point of a dodecahedron I thought about how steep a hill one would live on, and what that would look like perceptually, and how easy, if terrifyingly fast, if would be to bicycle over to a neighbors house (assuming a perfect bearings, lack of wind resistance, etc.)
I was actually thinking when I woke up this morning, "Gee, I really hope they discover something new about Dodecahedrons" and, well, there you have it. Could this day get any better?
Think not.
"...my colleagues and I have been exploring a _facet_ of what would it be like to live on a platonic solid"
Friendly I should think.
Very edgy. Did you get the point?
The Roman Dodecahedrons reminds me of the ball for different size yarn or cloth or rope we used as a child to make rugs of all types. Different size holes made for using different types of materials to make a long rope for rugs. The little buttons are perfect for what we used to use such as nails to help loop the yarn or such into making the weave for the rope of the rugs.
The dodecehedron has fascinated me recently and I was not sure why. I was learning mandala, and found a way to connect points on concentric circles to make an artistic rendition of a perfect dodecehedron. Interesting how you could both run in a straight line on both a sphere and the dodecehedron and end up In the same place.
How about 'Pathy McPathyface' for the shortest path's name
Don't be muscling in on my "Yellow Brick Road" idea! :)
What a time to be alive.
Came here to make this joke.
This is immediately where my mind went when I heard the speaker suggest looking for suggestions in the comments.
what about ‘P’
Starting my postgraduate studies in Mathematics this year and it's partly Numberphile's fault for featuring such well articulated topics. Hopefully one day I can articulate my ideas on the channel.
I feel like this is somehow going to improve battery technology.
It might improve teleporting technology in 2000 years. Who knows
Most of the theoretical physics in practical application is from maxwell's laws of electromagnetism and relativity which use math that is pretty standard for an undergraduate ,most of the pure math being worked on today ,ppl dont even know when and where it will be applied , mathematicians are stiving only to better the field so that when the time comes its ready for physicists
I have watched it second time after a break and I must say this scientist has ability to talk about maths with passion and involvement!
"The Dodequator"
The Mathematician‘s Dark Mark. 11:50
They wait when Dark Lord Euler will return.
"the Platonic Jogger"
Not only can you be a reclusive mathematician living on a dodecahedron... but you can also choose how long you want to run.
Lord Queezle Uh, I didn’t have time to watch till the end. Are you saying you could have an infinite run without passing through any of the corners? That is mind ... blown...
@@roygalaasen I haven't tried to prove it, but an infinitely long trajectory should be possible on any platonic solid. If you put the unfolded map on a coordinate plane, with at least two vertices on the x-axis, I conjecture that no trajectory with an irrational slope will intersect more than one vertex nor hit that vertex more than once (that is, it will intersect only one of the infinitely many colored points). It should be easiest to prove for the tetrahedron, since its map easily tesselates the plane. (If the map doesn't tesselate the plane, the slope might be different in different places because of the need to glue together sides that aren't touching, but it should always remain irrational.) A similar setup is often used to show how few rational numbers there are as compared to the irrational numbers around them.
Jeremy Davis yes, I think you are right. I realised that it should be possible short after posting my comment. Thank you for your explanation.
@@jeremydavis3631 It boils down to cardinality. The path is completely defined by the starting direction. There is a continuum of possible starting directions, while there are only countably many directions where you will eventually hit a vertex (that includes non-solutions due to running into a neighbor's house). Therefore most paths are infinite.
This guy should be a maths teacher...
You can see he wants to say so many things at the same time and he's digging for the right words that everyone could understand and he uses awesome examples ,🙏🙏
He is!
"There's the cube. Everyone knows the cube."
Scanlan? >.>
"Cube, glorious cube!"
I knew I'd find critters under a video on dodecahedrons
@@kane2056 How could we not be here in some form. :) This is actually really cool, I wonder what the applications of this will be.
Julian Wörner same, although i was expecting more c2 rather than c1 lol
another suggestion:
le petit path
So, path in French is chemin (according to Google translate), but another word is trajet, and I like that better because of its similarity to trajectory. Le Petit Trajet.
Is there anyway we can make this have 12 letters? Other than swapping le with the, since The Petit Path sounds awkward with the french version of the word in the middle of english.
Somehow i think La petite path is more appropriate because i assume the path is feminine (yep in french you assume gender for everything). But it really depends how you translate it. For example chemin and trajet are both masculine (le petit), but the first translation that came to my mind is trajectoire (la petite). Trajet and chemin are more like roads or itinerary, trajectoire is the word you would use in maths.
Yes
do "mon chemin quotidien" or something bain/pain if theres a better french pun
The code they did to solve this problem, is more impressive then the actual problem being solved, I just had a look, first thing I noticed was zero errors.😂
I'd buy a Numberphile shirt featuring that sheep with a rose in its mouth (aka, the Parker Sheep)
Perhaps you could do it on a Parker cube!
Wrong sort of dodecahedron for Matt
This man is explaining how get a Fragment of Possibility
Yyyyaaaaaahhhhhhhhhhhhhhhhhhhhh be pleased
Beep Beep
rewatched to enjoy some abstract thinking, but also to admire the narrator, Mr. Athreya's gentle voice
I love them all, but this was a particularly satisfying episode. Let's hear more from Jayadev and his colleagues!
Name suggestion for class 1: Lazy Prince Path
Name suggestion for class 31: Drunk Prince Path
LE: and have a spectrum from 1 to 31 going from Laziest to Drunkest
Which number would be the Ozzy path?
@@not2tired 32nd
It's weird how I have no interest in math in school, but I'm very interested in it while watching this channel.
High school math is boring
MATH: Magnificently Amazing, Taught Horribly
Stop this
I think it's because this channel only shows us the interesting conclusions, we aren't spending months grinding through the less sexy day-by-day work that it takes to get there. We like seeing finished products and concise explanations of them, but we don't want to be the one tasked with finding them. Think about how often presenters on this channel sort of gloss over "we can prove this but I won't do that here," because the machinations of why and how isn't as fun. Think of it like showing off game mechanics vs sitting for a year programming it.
Everything about this channel is amazing!!!!
Great video! I wonder if there is a relationship to the following:
Synergetics by R. Buckminster Fuller
457.40 Spherical Polyhedra in Icosahedral System: The 31 great circles of the spherical icosahedron provide spherical edges for three other polyhedra in addition to the icosahedron: the rhombic triacontrahedron, the octahedron, and the pentagonal dodecahedron. The edges of the spherical icosahedron are shown in heavy lines in the illustration.
Awesome! The animations and graphs are excellent. Love the way my mind is stretched with this channel. "The anti-social jogger problem" descriptor cracked me up.
One thing I don't understand is why this wasn't discoverd earlier, Since unfolding the shape and and drawing lines must have been tried before. What made it so hard?
Proving it
The proof - he mentioned this near the beginning.
Numberphile: platonic solids
Me and my crush: solidly platonic
Oof.
A flat picture hanging Laser and origami shape gives this nice perspective in the hand.
The use of a mirror makes it great to teach kids this kind of mind-blowing stuff .
They don't have to understand it to be intrigued ....
Energy follows these same patterns.
For old people in the US the most famous shortcut is the Slauson Cutoff. I'm fascinated with affine transforms. More please.
I’ve been building an imaginary world based on an icosahedron but I might consider a dodecahedron too now!
Why not a sphere?
Voltorb I’m a topologist, as he said in the video, they’re all the same in my eyes! But really it’s interesting to think about how the high ridges would affect how inhabitants experience gravity. By introducing an in universe magic it can be fun to explore how such a planet would come to exist and how it changes over time!
Voltorb arent spheres just boring orbs?
I love how Brady went in with the "I can make this outta cardboard!" and then Jayadev was all like, "Nah, let me move these goalposts a bit..."
The Stanley Parable Adventure Line! A perfect name for line with intersection 8.
Yes!! My mind instantly jumped to cutting and measuring the shapes too! Great video :D
Given the icosahedron and dodecahedron are duals of one another, I’m surprised the same property does not extend to the icosahedron.
There’s always something to discover even when the ideas run out.
18:20 - I was wondering if you could turn these areas into blocks and then turn them into a Rubik's Puzzle by mounting them on an internal moving sphere or system of rotating axes. I used to have such a puzzle called the "Skewb Ultimate", invented by Tony Durham, but that one only had 12 blocks/areas (same number as the amount of sides). I assume the number of blocks can be increased using these newly discovered paths, but maybe the arrangement of these blocks will prevent them from moving (i.e. rotating around an axis).
I would love to have some feedback on this from those who are familiar with this type of puzzle.
At first I was thinking the same, then I realized that (I think) it isn't possible.
To be able to turn these blocks you have to make the cut along one plane.
So basically you take the intersection of a plane and the dodecahedron.
That way you always end up with a path that never intersects itself.
And I think these paths that are created through intersection with a plane are not straight paths according definition explained in the video.
Sadly impossible. Take any puzzle cube (solid?) and rotate a side halfway. The edges of the plane of rotation form a closed path that excludes all other planes of rotation. So in other words, there is always a multitude of closed paths on a puzzle cube. The exception being (not much of) a puzzle cube with only two blocks. Wouldn't work for path 1 here either btw, because the plane it forms isn't flat.
It's sad because this would have been an incredible puzzle really. +10 for inspiration.
Try the megaminx. Or teraminx. The only way to increase it is to make it bigger
kosmonaut dan o BB by
omg... not only that it proves the motion that you explained, but also if you put each of those guys together you would create the space which could be filled with what Terrence Howard was speaking about which also creates that same kind of motion almost as if moving down the spine in circular motion just as the earth moves away from the sun... giant puzzle complete into creation that which you could recreate into building a whole new planet. Instead of building vehicles, you could create a building which also would align with the motion of the winds knowing how radical wind motion is and the stability that will be able to withstand earthquakes and it could be of any shape or size and completed to bend or move from place to place... wow. my mind just exploded... thank you.
I could listen to this guy all day
I became fascinated with dodecahedrons a few months ago when in a museum I found one of the many artifacts of that shape found mainly in gallo-roman sites, its use completely unknown. Viruses are also made from multiple pentagons constructing a dodecahedron, it's like the hexagon of the 3D world.
Thats quite interesting. Though im pretty sure viruses (what first came to my mind was bacteriophages) have icosahedron shape instead of dodecahedron.
@@C4pungMaster I think they might do both
Coincidentally those ancient Roman dodecahedra with knobs on the corners were for glove making, and the first piece of thread making the opening for the hand takes the same route as this path.
2:29 You had the opportunity of puting Minecraft grass noises and you lost it
*opens RUclips, sees the thumbnail for this video* A video about dodecahedrons posted 12 hours ago? I guess it was meant to be
this channel disappeared from my feed for more than a year, i'm glad it popped up again, gotta hit that bell icon!
Amazing work. I can already imagine how this could be applied to 3D graphics optimisation problems.
From Figure 2 at 15:05 if the pentagon side length is "t", then the shortest path is: sqrt(33.5 + 14.5*sqrt(5)) * t which is approximately 8.12 * t
Aggravatingly close to 5*phi. I had to do an exact calculation on my own to make sure because I was hoping we could be semi-boring and name it the 5*phi line.
For a path that crosses itself the most: The Brady Braid!
Yes!!!
"The Athreyan Cut"
"The Jayadev-ision"
"The Friend Zone"