๐งต Geometry Question
Anonymous at Tue, 2 Jul 2024 16:21:18 UTC No. 16264456
Does there exist a space-filling rhombic disphenoid?
๐๏ธ Anonymous at Tue, 2 Jul 2024 16:29:29 UTC No. 16264466
That's a tesseract from an angle.
Anonymous at Tue, 2 Jul 2024 17:45:47 UTC No. 16264577
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Anonymous at Tue, 2 Jul 2024 21:30:14 UTC No. 16264981
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Anonymous at Wed, 3 Jul 2024 03:30:01 UTC No. 16265349
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Anonymous at Wed, 3 Jul 2024 03:37:28 UTC No. 16265357
Maybe you should ask your mom about how my thrombic fills the space between her isotittie.
Anonymous at Wed, 3 Jul 2024 12:19:12 UTC No. 16265851
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Anonymous at Wed, 3 Jul 2024 17:51:43 UTC No. 16266292
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Anonymous at Wed, 3 Jul 2024 17:56:47 UTC No. 16266303
>>16264456
No.
Source? Divine intuition
Anonymous at Wed, 3 Jul 2024 18:01:20 UTC No. 16266308
Yes, any object is space filling.
What makes space filling curves special is that they're 1d, anything higher than 1d can obviously fill space by making it arbitrarily large.
Anonymous at Wed, 3 Jul 2024 18:28:44 UTC No. 16266343
>>16266308
Wrong: https://mathworld.wolfram.com/Space
Anonymous at Wed, 3 Jul 2024 21:50:43 UTC No. 16266708
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Anonymous at Thu, 4 Jul 2024 02:50:58 UTC No. 16267054
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Anonymous at Thu, 4 Jul 2024 11:29:02 UTC No. 16267484
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Anonymous at Thu, 4 Jul 2024 18:25:05 UTC No. 16267885
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Anonymous at Fri, 5 Jul 2024 01:40:11 UTC No. 16268325
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Anonymous at Fri, 5 Jul 2024 02:24:31 UTC No. 16268358
>>16264456
Here's one in 4-dimensions.
disphenoidal 288-cell
The Will Of God at Fri, 5 Jul 2024 03:17:20 UTC No. 16268390
SAFE SPACE
https://www.youtube.com/watch?v=TBv
Boards Of Canada at Fri, 5 Jul 2024 03:23:03 UTC No. 16268397
>>16264466
>Telephasic Workshop
https://www.youtube.com/watch?v=he8
Anonymous at Fri, 5 Jul 2024 12:31:08 UTC No. 16268883
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Anonymous at Fri, 5 Jul 2024 14:20:14 UTC No. 16268991
It's huge
Anonymous at Fri, 5 Jul 2024 20:28:03 UTC No. 16269438
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Anonymous at Sat, 6 Jul 2024 02:54:15 UTC No. 16269867
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Anonymous at Sat, 6 Jul 2024 14:12:05 UTC No. 16270317
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Anonymous at Sat, 6 Jul 2024 19:16:45 UTC No. 16270610
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Anonymous at Sun, 7 Jul 2024 13:16:44 UTC No. 16271455
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Anonymous at Sun, 7 Jul 2024 18:51:16 UTC No. 16271784
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Anonymous at Mon, 8 Jul 2024 17:00:25 UTC No. 16272967
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Anonymous at Tue, 9 Jul 2024 02:57:48 UTC No. 16273627
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Anonymous at Tue, 9 Jul 2024 17:42:29 UTC No. 16274496
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Anonymous at Wed, 10 Jul 2024 00:27:41 UTC No. 16275348
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Anonymous at Thu, 11 Jul 2024 03:47:03 UTC No. 16276871
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Anonymous at Fri, 12 Jul 2024 18:32:34 UTC No. 16279088
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Anonymous at Sat, 13 Jul 2024 12:36:46 UTC No. 16280201
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Anonymous at Sat, 13 Jul 2024 13:36:48 UTC No. 16280236
>>16264456
Our math isn't advanced enough to answer this question
Anonymous at Sat, 13 Jul 2024 15:05:32 UTC No. 16280310
>>16264466
Tesseracts have quadrilateral faces, moron
Anonymous at Sat, 13 Jul 2024 15:19:34 UTC No. 16280327
>>16264577
>>16264981
>>16265349
>>16265851
>>16266292
>>16266708
>>16267054
>>16267484
>>16267885
>>16268325
>>16268883
>>16269438
>>16269867
>>16270317
>>16270610
>>16271455
>>16271784
>>16272967
>>16273627
>>16274496
>>16275348
>>16276871
>>16279088
Reporting this thread for spam. 35 replies and ~25 of them are from the same person.
Anonymous at Sat, 13 Jul 2024 20:13:45 UTC No. 16280595
>>16280310
Maybe it's a 4D icosahedron?
Anonymous at Sat, 13 Jul 2024 20:14:34 UTC No. 16280601
>>16280327
I bumped this thread twice because I find it interesting, and nobody has made any interesting posts yet.
Anonymous at Sun, 14 Jul 2024 13:36:18 UTC No. 16281346
>>16280595
https://en.wikipedia.org/wiki/24-ce
๐๏ธ Anonymous at Sun, 14 Jul 2024 21:41:09 UTC No. 16281652
>>16281346
I was right. It's a 4D icosahedron. It also goes by the names of icositetrachoron and icosatetrahedroid.
Anonymous at Mon, 15 Jul 2024 18:33:02 UTC No. 16282671
>>16280327
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๐๏ธ Anonymous at Mon, 15 Jul 2024 18:36:04 UTC No. 16282677
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B at Mon, 15 Jul 2024 18:37:46 UTC No. 16282678
>>16282677
Ehm
๐๏ธ Anonymous at Tue, 16 Jul 2024 05:05:40 UTC No. 16283179
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ivfvg ynzoqncyhfwf at Tue, 16 Jul 2024 07:29:28 UTC No. 16283261
I'll just analyze the obvious way to try to do it: join edges of the same length together. In order for this to work, at each edge the angle between the two faces that meet there must be [math]2\pi/n[/math] for some [math]n \geq 3[/math]. So let's try to calculate the angles between the faces.
Line the x, y, and z axes up with the rotational axes of symmetry of the disphenoid, and let <a,b,c> be a normal vector to one of the faces. Then <a,-b,-c>, <-a,b,-c>, and <-a,-b,c> are normal vectors to the other faces. The possible angles between the normal vectors are [math]\cos^{-1}\left(\frac{a^2-b^2-
[math]\cos(\alpha) + \cos(\beta) + \cos(\gamma) = 1[/math]
where [math]\alpha[/math], [math]\beta[/math], and [math]\gamma[/math] are the angles between faces, each of which occurs at two opposite edges.
Now we just search for angles of the form [math]2\pi/n[/math] ([math]n \geq 3[/math]) that satisfy this equation. There is exactly one solution,
[math]\cos(2\pi/4) + \cos(2\pi/6) + \cos(2\pi/6) = 1[/math].
But this would give us a tetragonal disphenoid rather than a rhombic one.
I don't know whether or not there might be a more clever way of doing it that bypasses this issue.
Anonymous at Tue, 16 Jul 2024 16:55:38 UTC No. 16283717
>>16283261
So that's a no, then?
Anonymous at Wed, 17 Jul 2024 02:37:21 UTC No. 16284436
>>16283717
No under the stated assumption. I would guess it's no in general, but I don't have a proof of that.
Anonymous at Wed, 17 Jul 2024 18:06:38 UTC No. 16285320
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Anonymous at Thu, 18 Jul 2024 06:43:47 UTC No. 16286126
>>16264456
>Does there exist a space-filling rhombic disphenoid?
If I go with the definition, provided by >>16266343, then any object capable of tesselation would be "space-filling". Now, I would have expected the isosceles/regular tetrahedron to be a rhombic disphenoid that can fill space, but apparently it can't.
>>16283261
>I don't know whether or not there might be a more clever way of doing it that bypasses this issue.
I don't see how you adressed the issue of tesselation in the first place. What you did, was simply proving that only n = 3 satisfies the condition such that all normal vectors of some object could be expressed as mere variations of a single normal vector.
Anonymous at Thu, 18 Jul 2024 07:05:57 UTC No. 16286139
>>16286126
Frankly, for a single object to be space-filling, it would have to be either completely regular or of a specific shape that would still allow it to fill out space if by rotation alone. And in the latter case, there could be no more than 4 rhombic disphenoids forming another shape that would have to be regular.
Anonymous at Thu, 18 Jul 2024 09:15:43 UTC No. 16286498
>>16286126
Do you understand what the [math]2\pi/n[/math] bit is about?
Anonymous at Thu, 18 Jul 2024 13:03:16 UTC No. 16287122
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Anonymous at Thu, 18 Jul 2024 13:21:36 UTC No. 16287146
>>16286498
>Do you understand what the 2ฯ/n bit is about?
I forgot about that, quite frankly.
Anonymous at Fri, 19 Jul 2024 12:45:51 UTC No. 16288333
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Anonymous at Fri, 19 Jul 2024 19:15:59 UTC No. 16288710
>>16286139
Why would it have to be regular? It's well-known that non-regular polyhedra can be space filling. For example Escher's solid, or the triakis truncated tetrahedron.
๐๏ธ Anonymous at Sat, 20 Jul 2024 01:51:08 UTC No. 16289103
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๐๏ธ Anonymous at Sat, 20 Jul 2024 13:23:54 UTC No. 16289524
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Anonymous at Sat, 20 Jul 2024 14:09:39 UTC No. 16289577
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Anonymous at Sat, 20 Jul 2024 14:27:20 UTC No. 16289599
>>16286139
>Why would it have to be regular?
I haven't specifically called for the condition that they have to be regular in order to be space-filling. What I have done is the following: I have introduced a claim, a condition, that some assembly of non-regular disphenoids would have to form a regular disphenoid in order for the non-regular disphenoids to be space-filling. In two dimensions, the potential max number of space-filling non-regular polygons needed to assemble a regular space-filling polygon would be 6. I can't tell for three or more dimensions.
Anonymous at Sat, 20 Jul 2024 20:06:01 UTC No. 16290006
>>16289599
A regular disphenoid would be a Platonic tetrahedron, which is famously not space-filling. If your claim were true, then there couldn't be any space-filling disphenoids. Since space-filling disphenoids do exist (for example one with sides 6, 6, 4), we can conclude that your premise is false.
Anonymous at Sat, 20 Jul 2024 20:16:36 UTC No. 16290021
>>16290006
huh. you think you're smarter than me? think again, bucko.
Anonymous at Sat, 20 Jul 2024 22:05:02 UTC No. 16290151
>>16264456
Rhombic dodecahedra are known to work.
Anonymous at Sat, 20 Jul 2024 23:04:31 UTC No. 16290220
>>16290151
https://polytope.miraheze.org/wiki/
Anonymous at Sun, 21 Jul 2024 10:35:48 UTC No. 16290699
>>16264456
no, why do you wanna know?
Anonymous at Sun, 21 Jul 2024 17:06:00 UTC No. 16291013
>>16290699
Can you prove that, or are you making an educated guess?
Anonymous at Sun, 21 Jul 2024 19:56:45 UTC No. 16291307
>>16291013
not doing your homework
๐๏ธ Anonymous at Mon, 22 Jul 2024 00:50:16 UTC No. 16291686
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Anonymous at Mon, 22 Jul 2024 16:30:47 UTC No. 16292290
>>16291307
1) What class could this possibly be homework for?
2) Even if it was your answer would be about three weeks too late.
3) If you don't know the answer, you can just not post; no-one's making you reply.
Anonymous at Mon, 22 Jul 2024 16:46:21 UTC No. 16292306
>>16292290
>1) What class could this possibly be homework for?
Introduction into Algebraic geometry
>2) Even if it was your answer would be about three weeks too late.
You probably hoped someone would solve your exercises in time, but you failed...
>3) If you don't know the answer, you can just not post; no-one's making you reply.
No one's stopping me from replying either.
Uhmmm, did you just get pwned hard? I feel like yes, very much so...
๐๏ธ Anonymous at Tue, 23 Jul 2024 00:13:56 UTC No. 16292910
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๐๏ธ Anonymous at Tue, 23 Jul 2024 17:13:15 UTC No. 16293616
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