Growing a hyperdodecahedron
August 2, 2011 10:44 AM Subscribe
This short computer graphics animation presents the regular 120-cell: a four dimensional polytope composed of 120 dodecahedra and also known as the hyperdodecahedron or hecatonicosachoron.
Gian Marco Tedesco's animation was part of the MathFilm Festival 2008, which also included Dice - if you like it, you'll probably like Hitoshi Akayama's other animations - and the chilling documentary Attack of the Note Sheep.
Gian Marco Tedesco's animation was part of the MathFilm Festival 2008, which also included Dice - if you like it, you'll probably like Hitoshi Akayama's other animations - and the chilling documentary Attack of the Note Sheep.
Very cool. I watched it twice.
posted by Mental Wimp at 11:00 AM on August 2, 2011
posted by Mental Wimp at 11:00 AM on August 2, 2011
(Also: Holy shit that was done entirely in POV-Ray.)
Why are you surprised? It's complex geometry, but it's not very fancy rendering.
posted by anigbrowl at 11:03 AM on August 2, 2011
Why are you surprised? It's complex geometry, but it's not very fancy rendering.
posted by anigbrowl at 11:03 AM on August 2, 2011
That was awesome. Now is someone can just explain tesseracts in a way that makes sense in my brain...
posted by Navelgazer at 11:05 AM on August 2, 2011
posted by Navelgazer at 11:05 AM on August 2, 2011
The last time I toyed with it was almost fifteen years ago; I'm just surprised it's still around.
posted by griphus at 11:05 AM on August 2, 2011
posted by griphus at 11:05 AM on August 2, 2011
Wish there was room in my brain for a fourth dimension. I want to get it. Neat animation but I don't. Everything after 2 dimensions was simply 3-dimensional.
posted by rahnefan at 11:10 AM on August 2, 2011
posted by rahnefan at 11:10 AM on August 2, 2011
I am no math whiz. I don't get how this is "four dimensional."
Isn't it just a three-dimensional object made of three-dimensional objects? Is calling it four-dimensional just a way to handwave the obvious fact that it's not actually "regular"?
posted by Sys Rq at 11:13 AM on August 2, 2011
Isn't it just a three-dimensional object made of three-dimensional objects? Is calling it four-dimensional just a way to handwave the obvious fact that it's not actually "regular"?
posted by Sys Rq at 11:13 AM on August 2, 2011
The last time I toyed with it was almost fifteen years ago; I'm just surprised it's still around.
Oh, I see! It's very well established as a backend, still. If you use something like Blender or one of the other 3d modeling platforms, I think you can just point it at a Pov-Ray installation and have it run pretty much seamlessly. Last time I did any 3d was a couple of years ago, but the workflow is a snap nowadays.
posted by anigbrowl at 11:17 AM on August 2, 2011
Oh, I see! It's very well established as a backend, still. If you use something like Blender or one of the other 3d modeling platforms, I think you can just point it at a Pov-Ray installation and have it run pretty much seamlessly. Last time I did any 3d was a couple of years ago, but the workflow is a snap nowadays.
posted by anigbrowl at 11:17 AM on August 2, 2011
Now is someone can just explain tesseracts in a way that makes sense in my brain...
You see this string. And this insect.
posted by DU at 11:20 AM on August 2, 2011 [5 favorites]
You see this string. And this insect.
posted by DU at 11:20 AM on August 2, 2011 [5 favorites]
Sys Rq: " Isn't it just a three-dimensional object made of three-dimensional objects? Is calling it four-dimensional just a way to handwave the obvious fact that it's not actually "regular"?"
The problem is that you can't really show us a four-dimensional object. You can only show us its three-dimensional slices (and those are represented as two spatial dimensions and time, i.e. animation). In the same way, a two-dimensional being wouldn't really be able to comprehend a sphere. All it could see is a series of circles.
/Flatland
posted by Plutor at 11:22 AM on August 2, 2011
The problem is that you can't really show us a four-dimensional object. You can only show us its three-dimensional slices (and those are represented as two spatial dimensions and time, i.e. animation). In the same way, a two-dimensional being wouldn't really be able to comprehend a sphere. All it could see is a series of circles.
/Flatland
posted by Plutor at 11:22 AM on August 2, 2011
> Is calling it four-dimensional just a way to handwave the obvious fact that it's not actually "regular"?
I think the 3D object we see is something like the regular 4D object's "shadow", and it's deformed in the same way that the shadow of the single dodecahedron is.
posted by lucidium at 11:33 AM on August 2, 2011
I think the 3D object we see is something like the regular 4D object's "shadow", and it's deformed in the same way that the shadow of the single dodecahedron is.
posted by lucidium at 11:33 AM on August 2, 2011
The distortion is sort of analogous to drawing a "cube" in two dimensions like the picture below:
-----------
|\ /|
| \ / |
| \---/ |
| | | |
| | | |
| | | |
| /---\ |
| / \ |
|/ \|
-----------
(There are similar drawings of a dodecahedron in two dimensions, which of course come out distorted, but we've reached the limits of my ASCII art skills.)
posted by madcaptenor at 11:52 AM on August 2, 2011 [3 favorites]
-----------
|\ /|
| \ / |
| \---/ |
| | | |
| | | |
| | | |
| /---\ |
| / \ |
|/ \|
-----------
(There are similar drawings of a dodecahedron in two dimensions, which of course come out distorted, but we've reached the limits of my ASCII art skills.)
posted by madcaptenor at 11:52 AM on August 2, 2011 [3 favorites]
I wish they could make MIDI in 4-D. (Still, this was fun to watch.)
posted by not_on_display at 12:05 PM on August 2, 2011
posted by not_on_display at 12:05 PM on August 2, 2011
That was awesome. Now is someone can just explain tesseracts in a way that makes sense in my brain...
A tesseract is the three dimensional shape that is created when you unfold a four-dimensional hypercube; in the same way that this is the two-dimensional shape created when you unfold a three-dimensional cube.
posted by alby at 12:06 PM on August 2, 2011
A tesseract is the three dimensional shape that is created when you unfold a four-dimensional hypercube; in the same way that this is the two-dimensional shape created when you unfold a three-dimensional cube.
posted by alby at 12:06 PM on August 2, 2011
That was awesome. Now is someone can just explain tesseracts in a way that makes sense in my brain...
Madeleine L'Engle has got you covered.
posted by vverse23 at 12:36 PM on August 2, 2011
Madeleine L'Engle has got you covered.
posted by vverse23 at 12:36 PM on August 2, 2011
That was neat. I love higher-dimensional space.
Dimensions math, a set of (multilingual!) videos that eases you into 4-dimensional spaces, and from there to fractals and other pretty pictures. If you liked the main video you'll like this.
Visualizing 4 dimensions using color; maybe more intuitive (with a nice picture of a Klein bottle at the end, showing how it really isn't self-intersecting!)
A patent on a 4-dimensional UI.
4-dimensional Rubik's cube. 5-dimensional Rubik's cube. Explore the 120-cell.
Adanaxis, a 4-dimensional space shooter. Surprisingly playable.
posted by BungaDunga at 12:39 PM on August 2, 2011 [1 favorite]
Dimensions math, a set of (multilingual!) videos that eases you into 4-dimensional spaces, and from there to fractals and other pretty pictures. If you liked the main video you'll like this.
Visualizing 4 dimensions using color; maybe more intuitive (with a nice picture of a Klein bottle at the end, showing how it really isn't self-intersecting!)
A patent on a 4-dimensional UI.
4-dimensional Rubik's cube. 5-dimensional Rubik's cube. Explore the 120-cell.
Adanaxis, a 4-dimensional space shooter. Surprisingly playable.
posted by BungaDunga at 12:39 PM on August 2, 2011 [1 favorite]
More amazing to me is the fact that there is no such thing as a "five-dimensional dodecahedron". This one only goes up to four.
posted by erniepan at 12:42 PM on August 2, 2011
posted by erniepan at 12:42 PM on August 2, 2011
Aren't all objects that exist for longer than a nano-instant "four-dimensional", really ?
posted by genghis at 12:44 PM on August 2, 2011
posted by genghis at 12:44 PM on August 2, 2011
Huh. I guess maybe I just don't see the point of >3-dimesional "objects." Is there one? It all just sounds like mathematical drug-talk to me.
posted by Sys Rq at 2:14 PM on August 2, 2011
posted by Sys Rq at 2:14 PM on August 2, 2011
For those saying that this just looks like a 3D object, I think you may be overlooking the parts where the object sort of turns inside out and "engulfs" the viewer's point-of-view. At one point we're outside of it, and at another, we seem to be on the inside. You can roughly think of that as the effect of rotating/twisting the object through the 4th spatial dimension.
If you're having trouble visualizing what the whole 4D object "looks like," that's because, strictly speaking, you can't, given that we're (practically speaking) 3-dimension beings inhabiting a 3-dimensional
posted by treepour at 2:19 PM on August 2, 2011
If you're having trouble visualizing what the whole 4D object "looks like," that's because, strictly speaking, you can't, given that we're (practically speaking) 3-dimension beings inhabiting a 3-dimensional
posted by treepour at 2:19 PM on August 2, 2011
Huh. I guess maybe I just don't see the point of >3-dimesional "objects." Is there one? It all just sounds like mathematical drug-talk to me.
They're every bit as real as anything else mathematicians study. Have you ever seen the set of positive integers? No, you haven't, and that's about as concrete and well-established as mathematics gets. But if you watched that video, you have seen a 3-dimensional projection of the 120-cell.
posted by baf at 2:39 PM on August 2, 2011 [1 favorite]
They're every bit as real as anything else mathematicians study. Have you ever seen the set of positive integers? No, you haven't, and that's about as concrete and well-established as mathematics gets. But if you watched that video, you have seen a 3-dimensional projection of the 120-cell.
posted by baf at 2:39 PM on August 2, 2011 [1 favorite]
Huh. I guess maybe I just don't see the point of >3-dimesional "objects." Is there one?
Yes.
Nonmathematicians sometimes think that because the world is three-dimensional, our models should only be three-dimensional. But this only makes sense if you insist on using only the most direct possible correspondence between model and thing modelled — a "point" in the model has to represent a physical location in the world, a "space" in the model has to be a representation of the space we live in, and so on. Other relationships are possible and useful.
Consider, for example, a scatterplot of, I don't know, height versus weight. A point on this plot represents a certain combination of height and weight, which is an entirely abstract thing, not a physical point at all. A line on this plot represents a certain relation between height and weight, again an abstract thing, not corresponding to a physical line in any way. Of course it is useful to think of height/weight data arranged in this kind of abstract 2-dimensional "space".
The only difference between this height/weight scatterplot and a 25-dimensional scatterplot is the number of parameters in the data. From this point of view, it is pointless and absurd to impose the requirement that only 3-dimensional spaces be used.
(I do not claim any such practical application for the object considered in the post.)
(A 3-dimensional projection of the 120-cell is hanging from the ceiling of the Fields Institute.)
posted by stebulus at 2:43 PM on August 2, 2011 [5 favorites]
Yes.
Nonmathematicians sometimes think that because the world is three-dimensional, our models should only be three-dimensional. But this only makes sense if you insist on using only the most direct possible correspondence between model and thing modelled — a "point" in the model has to represent a physical location in the world, a "space" in the model has to be a representation of the space we live in, and so on. Other relationships are possible and useful.
Consider, for example, a scatterplot of, I don't know, height versus weight. A point on this plot represents a certain combination of height and weight, which is an entirely abstract thing, not a physical point at all. A line on this plot represents a certain relation between height and weight, again an abstract thing, not corresponding to a physical line in any way. Of course it is useful to think of height/weight data arranged in this kind of abstract 2-dimensional "space".
The only difference between this height/weight scatterplot and a 25-dimensional scatterplot is the number of parameters in the data. From this point of view, it is pointless and absurd to impose the requirement that only 3-dimensional spaces be used.
(I do not claim any such practical application for the object considered in the post.)
(A 3-dimensional projection of the 120-cell is hanging from the ceiling of the Fields Institute.)
posted by stebulus at 2:43 PM on August 2, 2011 [5 favorites]
This may be the set of convex figures the books were talking about in their math song?
posted by Buckt at 5:05 PM on August 2, 2011
posted by Buckt at 5:05 PM on August 2, 2011
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(Also: Holy shit that was done entirely in POV-Ray.)
posted by griphus at 10:51 AM on August 2, 2011 [7 favorites]