Klein diminishings

These are the 18 non-uniform CRF vertex diminishings of the quartic pentecontahexahedron (klein). There is one such uniform diminishing — the disantitridiminished pentecontahexahedron (datiduk). These, with a few exceptions, were all discovered by me using HyperRogue's experiment with geometry + creative modes by painting the surface of {7,3} on the Klein quartic. If you want to see a draft of this process to organize them, I wrote this text document in late July 2025: quarticdiminishings-2.txt. Due to all being manually discovered, there might very well be missing ones or even duplicates.

Naming

All of the names are composed of some sequence of location prefixes, the Latin numeral prefix of how many diminishings there are (for when there are at least two), the root adjective diminished and the operand, in this case the klein.

The distance prefixes are I. anti-, when the diminished vertex is opposite the root vertex, II. meta-, when the diminished vertex is two vertices away (lower angle) from the root vertex III. para-, when the diminished vertex is two vertices away (higher angle) from the root vertex.

The multiplicative prefixes are the Greek numeral adverbial prefixes, dis- for 2 and tris- for 3. They are used to indicate multiple of one type of locational diminishing.

The primary width prefixes are used to diambiguate arrangements when there are multiple of one type of locational diminishing, being steno- for a "thin" arrangement and plateo- for a "wide" arrangement.

Finally, the secondary width prefixes are used to diambiguate arrangements when there are multiple types of locational diminishings, being bracheo- for a "thin" arrangement and macro- for a "wide" arrangement. Essentially the multiple type equivalent of the primary width prefixes.

List

In the images below, heptagons are represented as groups of colored triangles. The central red heptagon is the "main" diminishing, the greenish heptagons are meta- diminishings, the bluish heptagons are para- diminishings and the yellow heptagon is an anti- diminishing.

Unidiminishing

• Diminished klein (duk) — has 7+14+7+7+7+7 triangular and 1 heptagonal faces. Has order 14 heptagonal symmetry. As the name implies, it is the diminishing of klein. Can be further diminished into abduk, mebduk and pabduk.

Bidiminishings

• Antibidiminished klein (A, abduk) — has 42_total triangular and 2 heptagonal faces. Has order 28 twice-heptagonal symmetry. Further diminishing of duk. Can be further diminished into amtiduk and datiduk.
• Metabidiminished klein (M, mebduk) — has 42_total triangular and 2 heptagonal faces. Has order 4 rectangular symmetry. Further diminishing of duk. Can be further diminished into amtiduk, dismetiduk, dipemtiduk and dippatiduk.
• Parabidiminished klein (P, pabduk) — has 42_total triangular and 2 heptagonal faces. Has order 4 rectangular symmetry. Further diminishing of duk. Can be further diminished into amtiduk, dismetiduk, dispatiduk and dippatiduk.

Tridiminishings

• Antimetatridiminished klein (AM, amtiduk) — has 35_total triangular and 1+1+1 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of abduk, mebduk and pabduk. Can be further diminished into adismequaduk and adipemquaduk.
• Distenometatridiminished klein (2sM, dismetiduk) — has 35_total triangular and 2+1 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of mebduk and pabduk. Can be further diminished into adismequaduk, badismepaquaduk, timquaduk and tipquaduk.
• Displateometatridiminished klein (2pM, dipemtiduk) — has 35_total triangular and 3 heptagonal faces. Has order 6 triangular symmetry. Further diminishing of mebduk. Can be further diminished into adipemquaduk and timquaduk.
• Distenoparatridiminished klein (2sP, dispatiduk) — has 35_total triangular and 3 heptagonal faces. Has order 6 triangular symmetry. Further diminishing of pabduk. Can be further diminished into adismequaduk and tipquaduk.
• Displateoparatridiminished klein (2pP, dippatiduk) — has 35_total triangular and 1+2 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of mebduk and pabduk.

Quadridiminishings

• Antidistenometaquadridiminished klein (A2sM, adismequaduk) — has 28_total triangular and 2+1+1 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of amtiduk, dismetiduk and dispatiduk. Can be further diminished into atimquiduk.
• Antidisplateometaquadridiminished klein (A2pM, adipemquaduk) — has 28_total triangular and 2+1+1 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of amtiduk, dipemtiduk and dippatiduk. Can be further diminished into atimquiduk.
• Macrometadisplateoparaquadridiminished klein (mM2pP, mamdippaquaduk) — has 28_total triangular and 4 heptagonal faces. Has order 8 twice-rectangular symmetry. Further diminishing of dippatiduk. Can be further diminished into dismedispaquiduk.
• Bracheodistenometaparaquadridiminished klein (b2sMP, badismepaquaduk) — has 28_total triangular and 2+2 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of dismetiduk and dippatiduk. Can be further diminished into dismedispaquiduk.
• Trismetaquadridiminished klein (3M, timquaduk) — has 28_total triangular and 2+2 heptagonal faces. Has order 4 rectangular symmetry. Further diminishing of dismetiduk and dipemtiduk. Can be further diminished into atimquiduk.
• Trisparaquadridiminished klein (3P, tipquaduk) — has 28_total triangular and 1+1+2 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of dismetiduk, dispatiduk and dippatiduk. Can be further diminished into atimquiduk and dismedispaquiduk.

Quinquediminishings

• Antitrismetaquinquediminished klein (A3M, atimquiduk) — has 21_total triangular and 1+2+1+1 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of adismequaduk, adipemquaduk, timquaduk, tipquaduk. It is a maximal diminishing.
• Distenometadistenoparaquinquediminished klein (2sM2sP, dismedispaquiduk) — has 21_total triangular and 1+2+2 heptagonal faces. Has order 2 bilateral symmetry. Further diminishing of mamdippaquaduk, badismepaquaduk and tipquaduk. Can be further diminished into dismetipasduk.

Sexadiminishing

• Distenometatrisparasexadiminished klein (2sMtP, dismetipasduk) — may simply be called the sexadiminished klein (seduk), has 2+6+6 triangular and 6 heptagonal faces. Has order 12 twice-triangular symmetry. Further diminishing of dismedispaquiduk. It is a maximal diminishing.