## Zalgaller's proof of 92 Johnson solids

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

### Zalgaller's proof of 92 Johnson solids

Quickfur, you can read Russian. What are the main ideas in Zalgaller's paper?

I think I can understand most of the paper, concerning trivalent and tetravalent vertices; I have been making similar pictures and calculations.

How does he handle pentavalent vertices, which have many degrees of freedom? I see some topological arguments involving the total numbers of elements of various types, and the graph formed by only-triangle vertices.

What is he doing in Figure 297? It reminds me of hyperbolic geometry, with equidistant curves and limiting-parallel lines.

Does he use the help of a computer? It looks like the work is mostly manual.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
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### Re: Zalgaller's proof of 92 Johnson solids

(I didn't know exactly how "bump" works, so I tested it on this topic. Original post was from April 8.)

Can anyone else read Russian? I guess quickfur doesn't have the time. Presumably he's interested, because these same ideas should be applicable to finding CRF polychora.

I'm not asking anyone to read through the whole paper in detail. I don't think it would take a lot of time to gather just enough information from it to answer some of my questions.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
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### Re: Zalgaller's proof of 92 Johnson solids

I'm pretty sure Zalgaller's paper has been translated into English somewhere. Maybe try searching on Google Scholar?

I've seen translations of Timofeenko's paper describing the algebraic coordinates of the 3D crown jewels; if that paper has been translated, which IMO is much less significant that Zalgaller's, then surely Zalgaller's paper should have been too.
quickfur
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