Minersphere::Index::Octolume of the E8 omnitruncate

The omnitruncated E₈ polyzetton, also known as the omnitruncated 4₂₁ polyzetton, is a gigantic eight dimensional polytope, in many aspects. As its name implies, it has the largest non-prismatic symmetry group in 8D, E₈, having order 696729600, and being its omnitruncate gives it the same number of vertices. It also consists of 881760 zetta of eight types. It also has the greatest circumradius, being √310 ≈ 17.60682 its edge length, of all non-prismatic uniform polytopes until 12D when it is beaten by the omnitruncated dodekeract (12-cube).^[1] This gives it a similarly massive octolume, which for a unit-edged version can be estimated by taking the octolume of an 8-ball of its circumradius, giving an upper bound of 57720062524τ⁴ ≈ 37483058817.005. However, the true octolume itself has never been calculated before to my knowledge. This page attempts to calculate it.

Terminology note: the dimension-specific supervolume terms here go volume -> quartolume -> quintolume -> sextolume -> septolume -> octolume etc.

Introduction

Calculating the supervolume

Calculating the supervolume of any polytope is quite simple, and uses very simple geometry. Specifically, the supervolume of a shape is simply the sum of supervolumes of the pyramids of the facets with the apex at the center of the polytope. Or, in more mathematical notation, V = ΣI_fA_fn⁻¹, with V being the total supervolume, I_f the inradius of a facet, A_f the superarea of a facet and n the dimension of the polytope.

Getting the superarea of the facets simply requires doing the same process, however getting the inradius of the facets requires the circumradius of the facet and of the full polytope, with I_f = √R²-r_f², with R being the polytope's circumradius and r_f the facet's circumradius. Getting the circumradius of any Wythoffian polytope is quite complicated and involves taking the inverse of its Coxeter diagram's Dynkin matrix.^[2] This is easy to do with code and has been calculated for all relevant polytopes here, so let's move on.

The omnitruncated E₈ polyzetton

The zetta of the E₈ omnitruncate can be determined using its Coxeter diagram, by removing singular nodes and their edges. This procedure for getting the facets is the same for any other polytope built using such diagrams. This gives the following eight types of zetta:

Count	Symmetry	Coxeter diagram	Name
240	E₇	x3x3x3x3x3x . *c3x	Omnitruncated E₇ polyexon
2160	D₇	. x3x3x3x3x3x *c3x	Great pentellated hecatonicosaexon
6720	E₆×A₁	x3x3x3x3x . x *c3x	Omnitruncated E₆ polypetal prism
17280	A₇	x3x3x3x3x3x3x .	Great petihexadecaexon
60480	D₅×A₂	x3x3x3x . x3x *c3x	Hexagonal-great runcinated triacontaditeral duoprism
69120	A₆×A₁	x . x3x3x3x3x *c3x	Great teritetradecapetal prism
241920	A₄×A₃	x3x3x . x3x3x *c3x	Truncated octahedral-great prismatodecachoral duoprism
483840	A₄×A₂×A₁	x3x . x3x3x3x *c3x	Hexagonal-great prismatodecachoral duoprismatic prism

Calculations

I first calculated its octolume semi-manually with a PARI/GP script (found here: E8.gp), which had a function that calculated the supervolume of a polytope using the polytope's Dynkin matrix and symmetry order along with the symmetry components' supervolume, circumradius and symmetry order. I call it "semi-manual" as it did require me to input all the polytope's compontents' properties individually. Its findings are listed below. As per standard, all of these were calculated with unit edge length.

Omnitruncated simplices

The formula for calculating the supervolume of omnitruncated simplices was already known, with $V_{otA n} = {(n + 1)}^{n - 1} \sqrt{\frac{n + 1}{2^{n}}}$ ,^[3] and was used to test if the numbers given by the script were accurate.

Name	Symmetry	Supervolume
Truncated octahedron	A₃	8√2 ≈ 11.313708
Great prismatodecachoron	A₄	1254√5 ≈ 69.877124
Great cellidodecateron	A₅	324√3 ≈ 561.184462
Great teritetradecapeton	A₆	168078√7 ≈ 5558.392786
Great petihexadecaexon	A₇	65536
Great exioctadecazetton	A₈	1434890716 = 896806.6875
Great zetticosayotton	A₉	6250000√5 ≈ 13975424.859374
Great yotticosidironnon	A₁₀	235794769132√11 ≈ 244388367.697871

The next omnitruncated simplex to have rational supervolume is the 17-simplex, with a septendecolume of 1423119505038213888. This follows sequence A155946 in the OEIS.

Omnitruncated demicubes

This category is equivalent to the subomnitruncated orthoplices. While the Polytope Wiki lists the volumes of these up to 6D, Klitzing's site only lists it for the 4D case.

Name	Symmetry	Supervolume
Truncated icositetrachoron	D₄	59
Great runcinated triacontaditeron	D₅	1968√2 ≈ 2783.172291
Great stericated hexecontatetrapeton	D₆	58835
Great pentellated hecatonicosoctaexon	D₇	1033896√2 ≈ 1462149.745283
Great hexicated diacosipentecontahexazetton	D₈	41822865
Great heptellated pentacosidodecayotton	D₉	957557536√2 ≈ 1354190854.163763
Great octicated chiliaicositetraronnon	D₁₀	48980938473

There is no known closed formula for these, although I do believe there probably is one.

Omnitruncated Gossets

This is the moment you have all been waiting for.^{[dubious—discuss]} They were all calculated on 29 September 2025 at around 00:08:20 UTC, when I was paying attention to the clock specifically to mention that fact. And yes, I did wait nine days before publishing this for that really nice release date of 8 October.

Name	Symmetry	Supervolume
Omnitruncated E₆ polypeton	E₆	111942√3 ≈ 193889.231501
Omnitruncated E₇ polyexon	E₇	31056795
Omnitruncated E₈ polyzetton	E₈	27386950905

Out of curiosity from having mentioned the latter's circumradius 8-ball octolume at the start, I thought of making a measure for "sphericity", defined as the ratio of a polytope's true supervolume to its circumradius supervolume, root its dimension ( $\sqrt[n]{\frac{V}{V_{B n} r^{n}}}$ ). Here are the sphericity values, as well as "smoothness" (= 11 − sphericity) for some other omnitruncates:

Name	Symmetry	Sphericity	Smoothness
Truncated octahedron	A₃	0.880783	8.388046
Great rhombicuboctahedron	B₃	0.928935	14.071570
Great rhombicosidodecahedron	H₃	0.964789	28.400523
Great prismatodecachoron	A₄	0.867523	7.548516
Great disprismatotesseractihexadecachoron	B₄	0.916658	11.998775
Great prismatotessaracontoctachoron	F₄	0.956230	22.846571
Great disprismatohexacosihecatonicosachoron	H₄	0.986144	72.170395
Omnitruncated E₆ polypeton	E₆	0.926496	13.604716
Omnitruncated E₇ polyexon	E₇	0.943000	17.543795
Omnitruncated E₈ polyzetton	E₈	0.961531	25.995394

General method

This section will now cover a more general method of calculation for any polytope with a respective Dynkin matrix. The code above uses this same method, but with some values input manually.

For a Dynkin matrix $𝐃_{n} = [\begin{matrix} 2 & - n_{1} & - n_{2} & \dots & - n_{n} \\ - n_{1} & 2 & - e_{12} & \dots & - e_{1 n} \\ - n_{2} & - e_{12} & 2 & \dots & - e_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ - n_{n} & - e_{1 n} & - e_{2 n} & \dots & 2 \end{matrix}]$ , its supervolume $V (𝐃_{n})$ can be calculated through this equation, recursing over all elements:

V (𝐃_{n}) = \frac{1}{n} \sum_{i = 1}^{n} \frac{O (𝐃_{n})}{O (𝐅_{i})} V (𝐅_{i}) \sqrt{R {(𝐃_{n})}^{2} - R {(𝐅_{i})}^{2}}

Here, $𝐅_{i}$ is the facet obtained from removing row and column $i + 1$ of $𝐃$ , corresponding to the facet obtained from removing node $i$ of the Coxeter diagram, $R (𝐃)$ is the circumradius of the polytope represented by $𝐃$ , equal to $1 - \frac{1}{2 {(𝐃^{- 1})}_{1,1}}$ , and $O (𝐃)$ is the order of the Coxeter group corresponding to $𝐃$ . This correspondence only considers the $e_{i j}$ values, with the Coxeter diagram edges being $f_{i j} = \frac{π}{\arccos \frac{e_{i j}}{2}}$ (or, inversely, $e_{i j} = 2 \cos \frac{π}{f_{i j}}$ ).

Getting the order of a Coxeter group from a Dynkin matrix is tricky. For convex cases above 2D, it is possible to get the group order by dividing the total hypersphere superarea by the superarea of a Schwarz simplex with the edge lengths being all values of $e_{i j}$ . For non-convex cases, it often results in non-integer order and dense polytopes, also explored in Failed Archimedeans.

One dimensional case

For a Dynkin matrix $𝐃_{1} = [\begin{matrix} 2 & - n_{1} \\ - n_{1} & 2 \end{matrix}]$ . The circumradius is simply $\frac{n_{1}}{2}$ , with the perimeter (1-volume) being $n_{1}$ . The order is also always 2. This is the trivial/base case.

Two dimensional case

For a Dynkin matrix $𝐃_{2} = [\begin{matrix} 2 & - n_{1} & - n_{2} \\ - n_{1} & 2 & - e_{12} \\ - n_{2} & - e_{12} & 2 \end{matrix}]$ , corresponding to the symmetry of the $f_{12}$ -gon, the circumradius $R$ becomes $\sqrt{\frac{n_{1}^{2} + n_{2}^{2} + e_{12} n_{1} n_{2}}{4 - e_{12}^{2}}}$ and the order $O$ becomes $2 f_{12} = \frac{2 π}{\arccos \frac{e_{12}}{2}}$ . The final area formula then becomes:

V (𝐃_{2}) = \frac{O}{4} (n_{1} \sqrt{R^{2} - \frac{n_{1}^{2}}{4}} + n_{2} \sqrt{R^{2} - \frac{n_{2}^{2}}{4}})

Three dimensional case

For polyhedra, the Dynkin matrix becomes $𝐃_{3} = [\begin{matrix} 2 & - n_{1} & - n_{2} & - n_{3} \\ - n_{1} & 2 & - e_{12} & - e_{13} \\ - n_{2} & - e_{12} & 2 & - e_{23} \\ - n_{3} & - e_{13} & - e_{23} & 2 \end{matrix}]$ . Calculations get more complicated in 3D with respect how the order and circumradius are calculated. Through the property that, for convex polyhedra, the sum of angular defects of vertices is equal to 4π, it is possible to obtain the order by taking the angular defect of the omnitruncate (summing the internal angles of the types of faces that make up the omnitruncate, and taking the reciprocal of the difference of that with 2π), resulting in $O (𝐃_{3}) = \frac{4 π}{2 π - α_{12} - α_{13} - α_{23}}$ , with $α_{i j} = π - \frac{π}{f_{i j}} = π - \arccos \frac{e_{i j}}{2}$ , which can be slightly reduced to $O (𝐃_{3}) = \frac{4}{β_{12} + β_{13} + β_{23} - 1}$ with $β_{i j} = \frac{1}{π} \arccos \frac{e_{i j}}{2}$ . In order to make calculating the circumradius easier, we'll first calculate $Q = {(𝐃_{3}^{- 1})}_{1,1}$ , which, again, $R (𝐃_{3}) = \sqrt{1 - \frac{1}{2 Q}}$ :

Q = \frac{- 2 e_{12}^{2} - 2 e_{13}^{2} - 2 e_{23}^{2} - e_{12} e_{13} e_{23} + 8}{(e_{23}^{2} - 4) n_{1}^{2} + (e_{13}^{2} - 4) n_{2}^{2} + (n_{3}^{2} - 4) e_{12}^{2} - 4 e_{13}^{2} - 4 e_{23}^{2} - 4 n_{3}^{2} + 2 (- n_{3} e_{12} e_{23} - 2 n_{3} e_{13}) n_{1} + 2 (- 2 e_{12} - e_{13} e_{23} - 2 e_{23} n_{3} - n_{3} e_{12} e_{13}) n_{2} - 4 e_{12} e_{13} e_{23} + 16}

The final volume formula then becomes:

V (𝐃_{3}) = \frac{O (𝐃_{3})}{6} (\frac{1}{f_{23}} V (𝐅_{1}) \sqrt{R {(𝐃_{3})}^{2} - R {(𝐅_{1})}^{2}} + \frac{1}{f_{13}} V (𝐅_{2}) \sqrt{R {(𝐃_{3})}^{2} - R {(𝐅_{2})}^{2}} + \frac{1}{f_{12}} V (𝐅_{3}) \sqrt{R {(𝐃_{3})}^{2} - R {(𝐅_{3})}^{2}})

Four dimensional case

For polychora, the Dynkin matrix becomes $𝐃_{4} = [\begin{matrix} 2 & - n_{1} & - n_{2} & - n_{3} & - n_{4} \\ - n_{1} & 2 & - e_{12} & - e_{13} & - e_{14} \\ - n_{2} & - e_{12} & 2 & - e_{23} & - e_{24} \\ - n_{3} & - e_{13} & - e_{23} & 2 & - e_{34} \\ - n_{4} & - e_{14} & - e_{24} & - e_{34} & 2 \end{matrix}]$ . Things become even more complex in this stage, for the same reasons as the 3D case: the equation for both the circumradius and order become extremely complicated. It's moreso for the order, which requires calculation of the volume of a tetrahedron on a glome (3-sphere), which is way more intricate and makes use of functions like dilogarithms.^[4] The circumradius is not as complicated, but attempting to compute it in PARI/GP as I had done for previous circumradius calculations took way too long. Safe to say, I would not be rewriting that long of an equation in LaTeX to display it here. At this point, it should just be easier to calculate with the main supervolume equation.

References

↑ MinersHaven. Polyhedra–polyvendeka ranking. Warning: that page is extremely large.
↑ Klitzing, Richard. Circumradius Calculator for Wythoffians.
↑ Klitzing, Richard. Analogs§Omnitruncated Simplex otS_n.
↑ Murakami, J.; Yano, M. On the Volume of a Hyperbolic and Spherical Tetrahedron.

Octolume of the E8 omnitruncate