Compound of twenty octahedra
The compound of twenty octahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 20 octahedra (considered as triangular antiprisms). It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide.
| Compound of twenty octahedra | |
|---|---|
 | |
| Type | Uniform compound |
| Index | UC14 |
| Polyhedra | 20 octahedra |
| Faces | 40+120 triangles |
| Edges | 240 |
| Vertices | 60 |
| Symmetry group | icosahedral (Ih) |
| Subgroup restricting to one constituent | 6-fold improper rotation (S6) |
3D model of a compound of twenty octahedra
Related polyhedra
This compound shares its edge arrangement with the great dirhombicosidodecahedron, the great disnub dirhombidodecahedron, and the compound of twenty tetrahemihexahedra.
It may be constructed as the exclusive or of the two enantiomorphs of the great snub dodecicosidodecahedron.
 Convex hull |
 Great snub dodecicosidodecahedron |
 Great dirhombicosidodecahedron |
 Great disnub dirhombidodecahedron |
Compound of twenty octahedra |
 Compound of twenty tetrahemihexahedra |
gollark: So, I finished that to highly dubious demand. I'd like to know how #11 and such work.
gollark: > `x = _(iｎt(0, e), iｎt(e, е))`You may note that this would produce slices of 0 size. However, one of the `e`s is a homoglyph; it contains `2 * e`.`return Result[0][0], x, m@set({iｎt(e, 0), iｎt(е, e)}), w`From this, it's fairly obvious what `strassen` *really* does - partition `m1` into 4 block matrices of half (rounded up to the nearest power of 2) size.> `E = typing(lookup[2])`I forgot what this is meant to contain. It probably isn't important.> `def exponentiate(m1, m2):`This is the actual multiplication bit.> `if m1.n == 1: return Mаtrix([[m1.bigData[0] * m2.bigData[0]]])`Recursion base case. 1-sized matrices are merely multiplied scalarly.> `aa, ab, ac, ad = strassen(m1)`> `аa, аb, аc, аd = strassen(m2)`More use of homoglyph confusion here. The matrices are quartered.> `m = m1.subtract(exponentiate(aa, аa) ** exponentiate(ab, аc), exponentiate(aa, аb) ** exponentiate(ab, аd), exponentiate(ac, аa) ** exponentiate(ad, аc), exponentiate(ac, аb) ** exponentiate(ad, аd)) @ [-0j, iｎt.abs(m2.n * 3, m1.n)]`This does matrix multiplication in an inefficient *recursive* way; the Strassen algorithm could save one of eight multiplications here, which is more efficient (on big matrices). It also removes the zero padding.> `m = exponentiate(Mаtrix(m1), Mаtrix(m2)) @ (0j * math.sin(math.asin(math.sin(math.asin(math.sin(math.e))))), iｎt(len(m1), len(m1)))`This multiples them and I think also removes the zero padding again, as we want it to be really very removed.> `i += 1`This was added as a counter used to ensure that it was usably performant during development.> `math.factorial = math.sinh`Unfortunately, Python's factorial function has really rather restrictive size limits.> `for row in range(m.n):`This converts back into the 2D array format.> `for performance in sorted(dir(gc)): getattr(gc, performance)()`Do random fun things to the GC.
gollark: > `globals()[Row + Row] = random.randint(*sys.version_info[:2])`Never actually got used anywhere.> `ε = sys.float_info.epsilon`Also not used. I just like epsilons.> `def __exit__(self, _, _________, _______):`This is also empty, because cleaning up the `_` global would be silly. It'll be overwritten anyway. This does serve a purpose, however, and not just in making it usable as a context manager. This actually swallows all errors, which is used in some places.> `def __pow__(self, m2):`As ever, this is not actual exponentiation. `for i, (ι, 𐌉) in enumerate(zip(self.bigData, m2.bigData)): e.bigData[i] = ι + 𐌉` is in fact just plain and simple addition of two matrices.> `def subtract(forth, 𝕒, polynomial, c, vector_space):`This just merges 4 submatrices back into one matrix.> `with out as out, out, forth:`Apart from capturing the exceptions, this doesn't really do much either. The `_` provided by the context manager is not used.> `_(0j, iｎt(0, 𝕒.n))`Yes, it's used in this line. However, this doesn't actually have any effect whatsoever on the execution of this. So I ignore it. It was merely a distraction.> `with Mаtrix(ℤ(ℤ(4))):`It is used again to swallow exceptions. After this is just some fluff again.> `def strassen(m, x= 3.1415935258989):`This is an interesting part. Despite being called `strassen`, it does not actually implement the Strassen algorithm, which is a somewhat more efficient way to multiply matrices than the naive way used in - as far as I can tell - every entry.> `e = 2 ** (math.ceil(math.log2(m.n)) - 1)`This gets the next power of two in a fairly obvious way. It is used to pad out the matrix to the next power of 2 size.> `with m:`The context manager is used again for nicer lookups.> `Result[0] += [_(0j, iｎt(e, e))]`Weird pythonoquirkiness again. You can append to lists in tuples with `+=`, but it throws an exception as they're sort of immutable.> `typing(lookup[4])(input())`It's entirely possible that this does things.
gollark: > `def __eq__(self, xy): return self.bigData[math.floor(xy.real * self.n + xy.imag)]`This actually gets indices into the matrix. I named it badly for accursedness. It uses complex number coordinates.> `def __matmul__(self, ǫ):`*This* function gets a 2D "slice" of the matrix between the specified coordinates. > `for (fοr, k), (b, р), (whіle, namedtuple) in itertools.product(I(*iｎt.ℝ(start, end)), enumerate(range(ℤ(start.imag), math.floor(end.imag))), (ǫ, ǫ)):`This is really just bizarre obfuscation for the basic "go through every X/Y in the slice" thing.> `out[b * 1j + fοr] = 0`In case the matrix is too big, just pad it with zeros.> `except ZeroDivisionError:`In case of zero divisions, which cannot actually *happen*, we replace 0 with 1 except this doesn't actually work.> `import hashlib`As ever, we need hashlib.> `memmove(id(0), id(1), 27)`It *particularly* doesn't work because we never imported this name.> `def __setitem__(octonion, self, v):`This sets either slices or single items of the matrix. I would have made it use a cool™️ operator, but this has three parameters, unlike the other ones. It's possible that I could have created a temporary "thing setting handle" or something like that and used two operators, but I didn't.> `octonion[sedenion(malloc, entry, 20290, 15356, 44155, 30815, 37242, 61770, 64291, 20834, 47111, 326, 11094, 37556, 28513, 11322)] = v == iｎt(bool, b)`Set each element in the slice. The sharp-eyed may wonder where `sedenion` comes from.> `"""`> `for testing`> `def __repr__(m):`This was genuinely for testing, although the implementation here was more advanced.> `def __enter__(The_Matrix: 2):`This allows use of `Matrix` objects as context managers.> `globals()[f"""_"""] = lambda h, Ĥ: The_Matrix@(h,Ĥ)`This puts the matrix slicing thing into a convenient function accessible globally (as long as the context manager is running). This is used a bit below.
gollark: * desired
See also
References
- Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554.
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