## All Johnson volumes found!

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### All Johnson volumes found!

I'd like to announce that as of today, we finally have exact algebraic expressions for the volumes of every single Johnson solid. Let me elaborate.

The volumes of J1–J83 are trivial to calculate in terms of the volumes of the Archimedean solids. The volumes of J84, J85, J86, J91, and J92 were already known by Wolfram|Alpha, and J87 is trivial to calculate in terms of J86. This leaves only J88–J90, which are the sphenomegacorona, the hebesphenomegacorona, and the disphenocingulum. I'll explain how I calculated the sphenomegacorona's volume – the other two can be done with the same method.

So first, we take its coordinates. Fortunately for us, these have already been calculated in [1] (and can be readily found in Wikipedia): we let k ≈ 0.59463 be the smallest positive root of the polynomial

​ ​ ​ ​ ​ ​ ​ ​ 1680 x¹⁶ – 4800 x¹⁵ – 3712 x¹⁴ + 17216 x¹³ + 1568 x¹² – 24576 x¹¹ + 2464 x¹⁰ + 17248 x⁹ – 3384 x⁸ – 5584 x⁷ + 2000 x⁶ + 240 x⁵ – 776 x⁴ + 304 x³ + 200 x² – 56 x – 23,

and we further define

​ ​ ​ ​ ​ ​ ​ ​ u = √(1 – k²), v = √(2 + 4k – 4k²), w = √(3 – 4k²).

The coordinates for a unit edge length sphenomegacorona are thus given by

• (0, ±1/2, u),
• k, ±1/2, 0),
• (0, ±(w + u)/(2u), (1 – 2k²)/(2u)),
• (±1/2, 0, –v/2),
• (0, ±(w(2k² – 1)/(2u³) + 1/2), (2k⁴ – 1)/(2u³)).
We can take advantage of the fact that J88 (as well as J89 and J90) can be divided into four congruent parts by two perpendicular planes. We'll calculate the volume of a single one of these parts via the following procedure:

1. We triangulate all of the portions of faces that lie in this sector.
2. We project all of the triangles onto one of the planes. This yields many "right truncated triangular prisms." Fortunately for us, their volume has a very simple formula, which is given and proven in this Math StackExchange answer.
3. We use a symbolic calculator to calculate the volumes of all of these triangular prisms in terms of the variables defined above, and add them all up.
At this point, we'll have our volume only in terms of many algebraic numbers:

​ ​ ​ ​ ​ ​ ​ ​ 1/12 (12 k u + v + 4 k v + 8 k w - ((1 + 2 k) (1 + 2 u⁴ + v w - 2 u² (2 + v w)))/u³)

Adding them up naively would be extremely expensive. Instead, we'll build up a Gröbner basis by taking the defining polynomial for k, along with the other polynomials that our variables must be roots of. In these case, these are:

• 1680 k¹⁶ – 4800 k¹⁵ – 3712 k¹⁴ + 17216 k¹³ + 1568 k¹² – 24576 k¹¹ + 2464 k¹⁰ + 17248 k⁹ – 3384 k⁸ – 5584 k⁷ + 2000 k⁶ + 240 k⁵ – 776 k⁴ + 304 k³ + 200 k² – 56 k – 23,
• u² + k² – 1
• v² + 4 k² – 4 k – 2,
• w² + 4 k² – 3,
• (12 k u + v + 4 k v + 8 k w– 12 V) u³ – (1 + 2 k) (1 + 2 u⁴ + v w – 2 u² (2 + v w)),
where V is the volume. By making V be sorted last, we can guarantee that the first entry of our Gröbner basis will be a polynomial that has V as a root. We can then factor our polynomial, and numerically check which of the factors has the actual volume as a root: this will be our minimal polynomial. By doing this, we conclude that the volume of a sphenomegacorona with unit edge length is the greatest real root (≈ 1.94811) of

​ ​ ​ ​ ​ ​ ​ ​ 521578814501447328359509917696 x³² – 985204427391622731345740955648 x³⁰ – 16645447351681991898880656015360 x²⁸ + 79710816694053483249372512649216 x²⁶ – 152195045391070538203422101864448 x²⁴ + 156280253448056209478031589244928 x²² – 96188116617075838858708654227456 x²⁰ + 30636368373570166303441645731840 x¹⁸ + 5828527077458909552923002273792 x¹⁶ – 8060049780765551057159394951168 x¹⁴ + 1018074792115156107372011716608 x¹² + 35220131544370794950945931264 x¹⁰ + 327511698517355918956755959808 x⁸ – 116978732884218191486738706432 x⁶ + 10231563774949176791703149568 x⁴ – 366323949299263261553952192 x² + 3071435678740442112675625.

For the hebesphenomegacorona, we get the greatest real root (≈ 2.91291) of

​ ​ ​ ​ ​ ​ ​ ​ 7330370277129322496 x²⁰ – 722445512980071186432 x¹⁸ + 3596480447590271287296 x¹⁶ – 8432333285523990773760 x¹⁴ + 8973584611317745975296 x¹² – 3065290664181478981632 x¹⁰ + 366229890219212144640 x⁸ – 8337259437908852736 x⁶ – 22211277300912896 x⁴ + 132615435213216 x² + 2693461945329.

Finally, for the disphenocingulum, we get the greatest real root (≈ 3.77765) of
​ ​ ​ ​ ​ ​ ​ ​ 1213025622610333925376 x²⁴ + 54451372392730545094656 x²² – 796837093078664749252608 x²⁰ – 4133410366404688544268288 x¹⁸ + 20902529024429842816303104 x¹⁶ – 133907540390420673677230080 x¹⁴ + 246234688242991598853881856 x¹² – 63327534106871321714442240 x¹⁰ + 14389309497459555704164608 x⁸ + 48042947402464500749392128 x ⁶ – 5891096640600351061013664 x⁴ – 3212114716816853362953264 x² + 479556973248657693884401.

References
Timofeenko, A. V. (2009). "The non-Platonic and non-Archimedean noncomposite polyhedra". Journal of Mathematical Science. 162 (5): 720.
I'm building a library of polyhedra and polychora: https://drive.google.com/drive/u/0/folders/1nQZ-QVVBfgYSck4pkZ7he0djF82T9MVy
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