The 6-orthoplex's dipetal angle is B = arccos(-4/6) = 131.8103°.
The 9-simplex's diyottal angle is C = arccos(1/9) = 83.6206°.
A = 180° - C/2 = 270° - BDoes this have any geometric substance? Or is it just an algebraic coincidence?
Relation between 3D icosahedron, 6D orthoplex, 9D simplex
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Relation between 3D icosahedron, 6D orthoplex, 9D simplex
by mr_e_man » Thu May 08, 2025 9:36 pm
The icosahedron's dihedral angle is A = arccos(-√5/3) = 138.1897°.
The 6-orthoplex's dipetal angle is B = arccos(-4/6) = 131.8103°.
The 9-simplex's diyottal angle is C = arccos(1/9) = 83.6206°.
A = 180° - C/2 = 270° - B
Does this have any geometric substance? Or is it just an algebraic coincidence?
The 6-orthoplex's dipetal angle is B = arccos(-4/6) = 131.8103°.
The 9-simplex's diyottal angle is C = arccos(1/9) = 83.6206°.
A = 180° - C/2 = 270° - BDoes this have any geometric substance? Or is it just an algebraic coincidence?
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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Re: Relation between 3D icosahedron, 6D orthoplex, 9D simple
by mr_e_man » Thu May 08, 2025 9:43 pm
Also:
The dodecahedron's dihedral angle is D = arccos(-√5/5) = 116.5651°.
The 5-orthoplex's diteral angle is E = arccos(-3/5) = 126.8699°.
D = 180° - E/2
The 600-choron's dichoral angle is F = arccos(-(1+3√5)/8) = 164.4775°.
The 5-choron's dichoral angle is G = arccos(1/4) = 75.5225°.
F = 240° - G
And all the other convex regular polytopes, except simplices and orthoplices, have rational ditopal angles.
And any simplex angle is half of an orthoplex angle, of squared dimension.
So I'm interested in finding relations between orthoplex angles. I found a few:
https://math.stackexchange.com/question ... ngle-60-de
The dodecahedron's dihedral angle is D = arccos(-√5/5) = 116.5651°.
The 5-orthoplex's diteral angle is E = arccos(-3/5) = 126.8699°.
D = 180° - E/2
The 600-choron's dichoral angle is F = arccos(-(1+3√5)/8) = 164.4775°.
The 5-choron's dichoral angle is G = arccos(1/4) = 75.5225°.
F = 240° - G
And all the other convex regular polytopes, except simplices and orthoplices, have rational ditopal angles.
And any simplex angle is half of an orthoplex angle, of squared dimension.
So I'm interested in finding relations between orthoplex angles. I found a few:
https://math.stackexchange.com/question ... ngle-60-de
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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