honeycomb

en-noun

honeycomb (countable and uncountable, plural honeycombs)

Alhambra
Oxford English Dictionary
United States Army

The noun is derived from Middle English hony comb, from Old English huniġcamb, from huniġ (“honey”) (ultimately from Proto-Indo-European *kn̥h₂ónks (“honey”)) + camb (“comb”) (ultimately from Proto-Indo-European *ǵómbʰos (“peg; tooth; row of teeth”)). The English word is analysable as honey + comb. The Oxford English Dictionary posits that the arrangement of several plates of wax “hanging parallel to each other from the roof of the hive suggests a comb with its teeth”. The verb is derived from the noun.

kennosto

something resembling honeycomb in having numerous cells or small holes

hunaja

something resembling honeycomb in sweetness; something desirable or pleasant

(countable, by extension) Something resembling honeycomb (sense 1) in having numerous cells or small holes.

The texture of the surface of a solar cell, intended to increase its surface area and capture more sunlight.

(chiefly aviation) Material manufactured with small hollow cells, sometimes sandwiched between two flat sheets, which is used to make light, stiff structural components.

kennosto

material manufactured with small hollow cells

A rudimentall reſemblance hereof there is in the cruciated and rugged folds of the Reticulum, or Net-like Ventricle of ruminating horned animals, vvhich is the ſecond in order, and culinarily called the Honey-comb.

[1658, Thomas Browne, “The Garden of Cyrus. […]. Chapter III.”, in Hydriotaphia, Urne-buriall, […] Together with The Garden of Cyrus, […], London: […] Hen[ry] Brome […], →OCLC, page 149:

Of the four ſtomachs vvith vvhich ruminant animals are furniſhed, the firſt is called the paunch, vvhich receives the food after it has been ſlightly chevved; the ſecond is called the honeycomb, and is properly nothing more than a continuation of the former; […]

[1791, Oliver Goldsmith, “Of Ruminating Animals”, in An History of the Earth, and Animated Nature. […], new edition, volume III, London: […] F[rancis] Wingrave, successor to Mr. [John] Nourse, […], →OCLC, page 3:

honeycomb stomach
honeycomb stomach#English

(countable, zoology) Short for honeycomb stomach (“the reticulum or second compartment of the stomach of a cow or other ruminant”).

honeycomb stomach

(uncountable, architecture) Short for honeycomb work (“especially in Moorish architecture: a form of ceiling ornamentation (especially of an arch or dome) made up of small vaulted arches”).

honeycomb work

honeycomb toffee

(countable, figurative) Something resembling honeycomb in sweetness; hence, something desirable or pleasant.

(uncountable, chiefly Australia, British, often attributive) A crumbly confection usually made by boiling together golden syrup, sugar, bicarbonate of soda, and water.


semiregular tessellation

en-noun

semiregular tessellation (plural semiregular tessellations)

(geometry) A tessellation of the plane by two or more different convex regular polygons.

Mathworld article on regular tessellation


regular tessellation

regular tessellation (plural regular tessellations)

(geometry) A tessellation of the plane by a convex regular polygon.


tessellate

en-verb

tessellate (third-person singular simple present tessellates, present participle tessellating, simple past and past participle tessellated)

tessellates (present singular third-person)
tessellating (participle present)
tessellated (participle past)
tessellated (past)

Etymology tree
Proto-Indo-European *kʷetwóres
Proto-Hellenic *kʷétwores
Ancient Greek τέσσαρες (téssares)bor.
Latin tessera
Proto-Indo-European *-lós
Proto-Italic *-elos
Latin -lus
Latin tessella
Latin -ātus
Latin tessellātuslbor.
English tessellate
Learned borrowing from Latin tessellātus, from tessella, diminutive of tessera; from Ancient Greek τέσσαρες (téssares), from Proto-Indo-European *kʷetwóres.

tessellieren

to completely fill with multiple copies of a two-dimensional shape edge to edge

Regular hexagons tessellate.

(intransitive, geometry) to cover a two-dimensional shape, such that multiple copies of itself placed edge to edge cover an area leaving no space between the shapes.


tessellation

en-noun

tessellation (countable and uncountable, plural tessellations)

Squares can be used for tessellation.

(uncountable) The property or fact of tessellating.


quasicrystal

(physics, chemistry) any solid with conventional crystalline properties but exhibiting a point group symmetry inconsistent with translational periodicity


quadrille

From French quadrillé (“crosshatched”, adjective).

bor

Quadrille ruled graph paper, quad paper.


tiling

en-noun

tiling (countable and uncountable, plural tilings)

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(geometry) A tessellation; the covering of a plane with shapes, without overlaps or gaps.


prototile

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(geometry) Any of the minimal set of non-congruent shapes from which a larger set of shapes, some of which may be congruent, can be produced.


quasiregular

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The element a can be uniquely represented in the form r + t, where [rt] = 0, t is nilpotent and r is a quasiregular element of G ([1]; p. 108).

1966, Translations: Lie groups, American Mathematical Society, Series 1, Volume 9, page 215:

A onesided ideal is quasiregular provided that it consists of quasiregular elements.

1976, Carl Faith, Algebra II: Ring Theory, page 33:

A norm #92;begin#123;Vmatrix#125;#92;end#123;Vmatrix#125; is called spectral if the group of quasiregular elements is open in the associated norm topology, equivalently if it satisfies Gelfand's spectral radius formula:[…].

1989, Proceedings of the Royal Irish Academy: Mathematical and physical sciences, page 128:

(mathematics, ring theory, of an element r of a ring) Such that 1 − r is a unit (has a multiplicative inverse).

Given two orientable Riemannian manifolds V₁ and V₂, one may ask whether a non-constant quasiregular map 𝑓 : V₁ → V₂ exists.

1999, Israel Mathematical Conference Proceedings, volumes 12-14, page 61:

1999, S. Mueller, Variational models for microstructure and phase transitions, F. Bethuel, G. Huisken, S. Mueller, K. Steffen (editors),Calculus of Variations and Geometric Evolution Problems, Springer, Lecture Notes in Mathematics, Volume 1713, page 108,
An alternative proof that features an interesting connection with the theory of quasiconformal (or more precisely quasiregular) maps proceeds as follows.

Quasiregular maps are a generalization of quasiconformal maps where the assumption of injectivity is relaxed. Heinonen and Holopainen [138] developed nonlinear potential theory and quasiregular maps on Carnot groups.

2007, Luca Capogna, Donatella Danielli, Scott D. Pauls, Jeremy Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, page 142:

mathematical analysis
mathematical analysis

(mathematics, mathematical analysis, of a mapping from a multi-dimensional space or manifold to an equivalent space) Having certain properties in common with holomorphic functions of a single complex variable.

in the fundamental affine space H = G/Z is called quasiregular.

1988, I. M. Gelfand, M. I. Graev, 3: Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry, Izrail M. Gelfand, Collected Papers, Volume II, page 357, We first decompose the quasiregular representations of a complex semi-simple Lie group into irreducible ones. A representation of the group G given by the formula T_g𝑓(h) = 𝑓(hg)

1998, Vladimir F. Molchanov, Discrete series and analyticity, Joachim Hilgert, Jimmie D. Lawson, Karl-Hermann Leeb, Ernest B. Vinberg (editors), Positivity in Lie Theory: Open Problems,De Gruyter Expositions in Mathematics, Volume 26, page 188,
As it is known (see [11], [13], [21], [22]), the quasiregular representation on the hyperboloid decomposes into two series of irreducible unitary representations: continuous and discrete.

2008, André Unterberger, Alternative Pseudodifferential Analysis: With an Application to Modular Forms, Springer, Lecture Notes in Mathematics, Volume 1935, page 6,
Note that the representation Met⁽²⁾, contrary to the quasiregular representation of the same group, does not act by changes of coordinates only […] .

(mathematics, representation theory, topological algebra, of a representation) That is the result of a required adjustment of an induced representation that would, unadjusted, give rise to (only) a quasi-invariant measure.

representation theory; topological algebra; representation theory; topological algebra


honeycomb conjecture

(mathematics) The proven conjecture that a regular hexagonal grid, or honeycomb, is the best way to divide a surface into regions of equal area with the smallest total perimeter.


hosohedron

A tessellation of lunes on a spherical surface, such that each lune shares the same two vertices.


monohedral

(mathematics, of a tessellation) Having exactly one prototile.


isotoxal

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(geometry, of a polytope or tiling) That has symmetries which act transitively on its edges.

edge-symmetric

having symmetries that act transitively on its edges


Schläfli symbol

Schläfli symbol (plural Schläfli symbols)

From Schläfli (“a surname”) + symbol, after 19th-century Swiss mathematician Ludwig Schläfli (1814–1895).

1995, Unnamed translator, E. B. Vinberg, Polynomial Group, article in M. Hazewinkel (editor), Encyclopaedia of Mathematics, Volume 4: Monge–Ampère Equation — Rings and Algebras, 2013, International Edition, page 481,
The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the tetrahedron — {3, 3}, the cube — {4, 3}, the octahedron — {3, 4} — the dodecahedron — {5, 3}, and the icosahedron — {3, 5}.

For the regular tiling of squares the Schläfli symbol is {4,4}, indicating that there are four squares surrounding each vertex. And for the regular tiling of hexagons the Schläfli symbol is {6,3}, indicating that there are three hexagons surrounding each vertex.

2007, Michael E. Mortenson, chapter L, in Geometric Transformations for 3D Modeling, page 248:

The 24-cell has has 24 octahedral cells, 24 vertices, 96 triangular faces and 96 edges. Its Schläfli symbol is {3, 4, 3} from which we see that the vertex figure is a cube.

2012, John Barnes, Gems of Geometry, 2nd edition, page 79:

(geometry) A notation that recursively encodes certain properties of a specified regular polytope or tessellation.

Schläflin symboli

notation encoding certain properties of a regular polytope or tessellation


hyperhoneycomb

Simple examples are hyperhoneycomb and stripy-honeycomb lattices, which are realized in #92;beta-Li#95;#123;2#125;IrO#95;#123;3#125; and #92;gamma-Li#95;#123;2#125;IrO#95;#123;3#125;, respectively.

2015, Motohiko Ezawa, “Loop-nodal and Point-nodal Semimetals in Three-dimensional Honeycomb Lattices”, in arXiv:

(mathematics) A three-dimensional analog of a honeycomb structure


Wang tile

First proposed by mathematician, logician, and philosopher Hao Wang in 1961.

(mathematics) A square tile with a colour on each side. Possible sets of such tiles are studied to determine whether they can tile the plane (arranged with like colours touching, and without rotations or reflections) and whether this can be done in a periodic pattern.


girih

From Classical Persian گره (giri, “knot”).

decorative Islamic geometric patterns used in architecture and handicrafts, consisting of angled lines that form an interlaced strapwork


girih tile

Any of a set of five tiles used in the creation of girih patterns.


mucube

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(geometry) A regular skew apeirohedron with six squares around each vertex, formed by an infinite number of cubic cells (specifically, cubes missing two opposite faces each, around cubic empty spaces with no faces) in Euclidean 3D space.


muoctahedron

muoctahedron (plural muoctahedra or muoctahedrons)

muoctahedra (plural)
muoctahedrons (plural)

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truncated octahedra
truncated octahedron

(geometry) A regular skew apeirohedron with four hexagons around each vertex, formed by an infinite number of octahedron-like cells (specifically, truncated octahedra missing their square faces, and joined by the resulting square-shaped holes) in Euclidean 3D space.

六角四片四角孔ねじれ正多面体

ろっかくしへんしかくこうねじれせいためんたい


mutetrahedron

mutetrahedron (plural mutetrahedra or mutetrahedrons)

mutetrahedra (plural)
mutetrahedrons (plural)

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truncated tetrahedron
truncated tetrahedron

(geometry) A regular skew apeirohedron with six hexagons around each vertex, formed by an infinite number of truncated tetrahedron-like cells (specifically, truncated tetrahedra missing their triangle faces, with the resulting triangle-shaped holes joined to form empty spaces in the shape of faceless tetrahedra) in Euclidean 3D space.

六角六片三角孔ねじれ正多面体

ろっかくろくへんさんかくこうねじれせいためんたい


Schläfli notation

Schläfli notation (plural Schläfli notations)


wallpaper group

wallpaper group (plural wallpaper groups)

(geometry) A mathematical group that classifies two-dimensional repeating patterns based on their symmetries.


planigon

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(geometry) A convex polygon that can fill the plane with only copies of itself.


monohedrally

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The extent of our ignorance is further illustrated by the fact that we do not know all the different shapes of convex pentagons which will tile the plane monohedrally.

1987, Branko Grünbaum, G[eoffrey] C[olin] Shephard, Tilings and Patterns, New York, N.Y.: W. H. Freeman and Company, →ISBN, page 22:

(geometry) In a monohedral manner; involving only copies of the original tile (allowing for translation, rotation, and reflection).


monotile

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In the following section we will describe families of tilings by a curious three-dimensional aperiodic monotile whose matching rule — for some values of its parameter — is weak but not strong.

1995, Marjorie Senechal, Quasicrystals and Geometry, Cambridge, Cambridgeshire: Cambridge University Press, →ISBN, page 208:

And I am sorry I never delivered on the four-dimensional Poultry design. It turns out John Horton Conway found four-dimensional and five-dimensional aperiodic monotiles sixty years ago, but it's not too well documented.

1997, Rudy Rucker, Freeware (Ware Tetralogy; 3), New York, N.Y.: Avon Books, →ISBN, page 69:

This naturally led to serious rumination about the possible existence of a single tile, or monotile, that forces non-periodic global structure.

2007 March, Joshua E. S. Socolar, “More ways to tile with only one shape polygon”, in The Mathematical Intelligencer, volume 29, number 2, New York, N.Y.: Springer, →DOI, →ISSN, →OCLC, page 33:

An aperiodic monotile never repeats a formation, no matter how long the pattern.

2023 April 4, Matthew Cantor, “'The miracle that disrupts order': mathematicians invent new 'einstein' shape”, in The Guardian, London: Guardian News & Media, →ISSN, →OCLC, archived from the original on 2023-04-04:

(geometry) A prototile tiling monohedrally; any shape able to completely tile some space on its own (allowing for translation, rotation, and reflection).