## Dimensional terms for ditopal angles in sub-polytopes

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### Dimensional terms for ditopal angles in sub-polytopes

Consider the angle between two k-dimensional elements in an n-dimensional polytope, both contained in a (k+1)-dimensional element, and both containing a (k-1)-dimensional element. We're not assuming that k=n-1.

Based on Wendy's terminology, the names for these would be as follows.
k=2: dihedral angle
k=3: dichoral angle
k=4: diteral angle
k=5: dipetal angle
k=6: diectal angle
k=7: dizettal angle
k=8: diyottal angle

What would it be for k=1?
dilatral?
dilateral?
bilateral?

digrammal?

Note that "-lateral" is taken from Latin, while "-gram" is taken from Greek, just like the others.

And "digonal" is inappropriate, since it may suggest some relation to digons, and anyway "-gon" refers to a vertex or angle of the polygon, not an edge of the polygon as intended.

(In fact, we might condense the above terms using "-gon" instead of "angle".
k=1: digrammagon
k=2: dihedragon
k=3: dichoragon
k=4: diteragon
k=5: dipetagon
k=6: diectagon
k=7: dizettagon
k=8: diyottagon
Or maybe not.)

What do you think?
Last edited by mr_e_man on Tue Sep 13, 2022 2:06 am, edited 1 time in total.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
Tetronian

Posts: 404
Joined: Tue Sep 18, 2018 4:10 am

### Re: Dimensional terms for ditopal angles in sub-polytopes

And what about the [k-1, k+1]-subpolytope structure itself?

I've called it a k-dyad, but it would be good to have names without numbers or variables mixed in.

ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
Tetronian

Posts: 404
Joined: Tue Sep 18, 2018 4:10 am

### Re: Dimensional terms for ditopal angles in sub-polytopes

Wendy’s term for this is ‘margin angle’.

Angle is read as the fraction of space occupied at the named surtope.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger

wendy
Pentonian

Posts: 1996
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

### Re: Dimensional terms for ditopal angles in sub-polytopes

Yes, I knew that term. It only applies to k=n-1, so it's hardly relevant, I thought.

"Ditopal angle" and "margin angle" are synonyms. But the interpretation is slightly different; "margin" refers to the (n-2)D thing, while "ditopal" refers to the two (n-1)D things.

The 24-choron has three different angles (here I'm dealing with 2D/circular angles, not solid angles). What I'm tentatively calling the digrammal angle is 60°; the dihedral angle is 109.4712°; and the dichoral angle is 120°.

We can't just call them all "margin angle". There's no point to using names unless the names are distinct.

It does happen often, though: A father and son may have the same name; then we use "senior" and "junior" to distinguish.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
Tetronian

Posts: 404
Joined: Tue Sep 18, 2018 4:10 am

### Re: Dimensional terms for ditopal angles in sub-polytopes

I'm not particularly sure this is a good idea, since in general, something like the dilateral angle would vary from hedron to hedron in any polytope, and would not particularly describe something unique to the global polytope in question.

For example, x3x4o has hexagons and squares, but the dilateral angle would simply be the margin-angles of the hexagons and squares by themselves, and could be explained such. margin-angle is already a localised down word, so the angle here is simply the margin-angle of the hexagon and square. Note this is a possesive construction, which means that it is a property of these figures.

If you has a locative construction (margin angle at the hexagon), then the general figure has several different margin angles, one is over a hexagon. For example, the octagonny o3x4x3o has two kinds of margin. The first is the octagon, over or at which the margin-angle is 135°, while over the triangles, it is 120°.

Note here that fractions 0:xx are given in successive columns of 120, or base 120. They are expanded to fractions in decimal numbers. The unit in every case is the full sphere, since this is preserved over a section, while radian-measure etc isn't.

Instead, i find the various solid angles useful. Note here, that if these angles are measured at the interior of some surtope, it is the same as the solid angle in the space orthogonal to the surtope. Then it's a matter of simply associating the solid fraction against the full space around a point. So for example, the hedric angle or margin angle of {3,4,3} is 1/3 (0:40). The choric or edge angle of the same is 0:30, or 1/4. The teric or vertex-angle is 0:15, or 1/8. These same fractions apply to the prisms that these form in higher dimensions. We see already that the edge-angle derives from the choric angle by the choric angle being the half-spherical excess, whether this is applied in 3d or 4d. Also, we see that for a convex solid, the angles decrease as the dimensions increase. For {5,3,3}, it's 0:48, 0:42, and 0:38.24 or 2/5, 7/20 and 191/600.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger

wendy
Pentonian

Posts: 1996
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia