Polyiamond

Polyiamond

A polyiamond (also polyamond or simply iamond) is a polyform whose base form is an equilateral triangle. The word polyiamond is a back-formation from diamond, because this word is often used to describe the shape of a pair of equilateral triangles placed base to base, and the initial "di-" looked like a Greek prefix meaning "two-".

Contents

Counting polyiamonds

The basic combinatorial question is how many different polyiamonds exist with a given number of cells. Like polyominoes, polyiamonds may be free or one-sided. Free polyiamonds are invariant under reflection as well as translation and rotation. One-sided polyiamonds distinguish reflections.

The number of free n-iamonds for n = 1, 2, 3, … is (sequence A000577 in OEIS):

1, 1, 1, 3, 4, 12, 24, 66, 160, …

The number of free polyiamonds with holes is given by OEISA070764; the number of free polyiamonds without holes is given by OEISA070765; the number of fixed polyiamonds is given by OEISA001420; the number of one-sided polyiamonds is given by OEISA006534.

Name Number of forms Forms
Moniamond 1
Polyiamond-1-1.svg
Diamond 1
Polyiamond-2-1.svg
Triamond 1
Polyiamond-3-1.svg
Tetriamond 3
Polyiamond-4-2.svg Polyiamond-4-1.svg Polyiamond-4-3.svg
Pentiamond 4
Polyiamond-5-1.svg Polyiamond-5-2.svg Polyiamond-5-3.svg Polyiamond-5-4.svg
Hexiamond 12
Polyiamond-6-1.svg Polyiamond-6-2.svg Polyiamond-6-3.svg Polyiamond-6-4.svg Polyiamond-6-5.svg Polyiamond-6-6.svg Polyiamond-6-7.svg Polyiamond-6-8.svg Polyiamond-6-9.svg Polyiamond-6-10.svg Polyiamond-6-11.svg Polyiamond-6-12.svg

Symmetries

Possible symmetries are mirror symmetry, 2-, 3-, and 6-fold rotational symmetry, and each combined with mirror symmetry.

2-fold rotational symmetry with and without mirror symmetry requires at least 2 and 4 triangles, respectively. 6-fold rotational symmetry with and without mirror symmetry requires at least 6 and 18 triangles, respectively. Asymmetry requires at least 5 triangles. 3-fold rotational symmetry without mirror symmetry requires at least 7 triangles.

In the case of only mirror symmetry we can distinguish having the symmetry axis aligned with the grid or rotated 30° (requires at least 4 and 3 triangles, respectively); ditto for 3-fold rotational symmetry, combined with mirror symmetry (requires at least 18 and 1 triangles, respectively).

Polyiamond-5-2.svg Polyiamond-4-1 (rotated).svg Polyiamond-3-1.svg Polyiamond-4-2.svg Polyiamond-2-1 (rotated).svg
Asymmetric Mirror, 0° Mirror, 30° Rotational, 2-Fold Mirror, 2-Fold
Polyiamond 3-fold rotational symmetry.svg Polyiamond 3-fold mirror symmetry (0 deg).svg Polyiamond-1-1.svg Polyiamond 6-fold rotational symmetry.svg Polyiamond-6-11.svg
Rotational, 3-Fold Mirror, 0°, 3-fold Mirror, 30°, 3-fold Rotational, 6-Fold Mirror, 6-Fold

Generalizations

Like polyominoes, but unlike polyhexes, polyiamonds have three-dimensional counterparts, formed by aggregating tetrahedra. However, polytetrahedra do not tile 3-space in the way polyiamonds can tile 2-space.

Tessellations

Every polyiamond of order 6 or less tiles the plane. All but one of the heptiamonds tile the plane.[1]

Correspondence with Polyhexes

Pentiamond with corresponding pentahex superimposed.

Every polyiamond corresponds to a polyhex, as illustrated at right. Conversely, every polyhex is also a polyiamond, because each hexagonal cell of a polyhex is the union of six adjacent equilateral triangles.

See also

External links

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • polyiamond — noun A polyform made by joining one or more equilateral triangles edge to edge in various arrangements. Syn: n iamond See Also: diamond …   Wiktionary

  • Полиамонд — Полиамонд  геометрическая фигура в виде многоугольника, составленного из нескольких одинаковых равносторонних треугольников. Наряду с полимино, широко распространена в занимательной математике, в основном в задачах на составление фигур.… …   Википедия

  • n-iamond — noun A polyiamond made up of n triangles. Syn: polyiamond …   Wiktionary

  • Polyomino — The 18 one sided pentominoes, including 6 mirrored pairs …   Wikipedia

  • Tessellation — A tessellation of pavement A honeycomb is an example of a t …   Wikipedia

  • List of combinatorics topics — This is a list of combinatorics topics.A few decades ago it might have been said that combinatorics is little more than a way to classify poorly understood problems, and some standard remedies. Great progress has been made since 1960.This page is …   Wikipedia

  • Diamond (disambiguation) — Diamond is the hardest known natural material. Diamond, ♦ or ◊ may also refer to: Diamond (gemstone) Diamonds, a suit of cards Alt codes, ♦, the character or symbol formed by using the Alt key in conjunction with the Number Pad (Alt+num4) Lozenge …   Wikipedia

  • Polyhex (mathematics) — In recreational mathematics, a polyhex is a polyform with a regular hexagon (or hex for short) as the base form.As with polyominoes, polyhexes may be enumerated as free polyhexes (where rotations and reflections count as the same shape), fixed… …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • Octomino — The 369 free octominoes An octomino (or 8 omino) is a polyomino of order 8, that is, a polygon in the plane made of 8 equal sized squares connected edge to edge.[1] The name of this type of figure is formed with the prefix …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”