Definitions of Terms

Animal
Any of the puzzle pieces that make up a set of Polyforms. Each Heptiamond is an animal and so is each Pentomino, etc. On this website, however, ‘puzzle piece’ is the standard name, for easy reading.

Bar

Bar
An elongated hexagonal Shape with smooth edges. A Wiener is a shorter Bar.

Barrels and Minibarrels,
Whole and Split

Barrel
A compact hexagonal Shape with smooth edges. There also exists an Isomorphic Subdivision into four Minibarrels. Barrels can be Whole or Split (in halves).

cell
on grid

Cell
The smallest space defined by intersecting gridlines on a plane. In the most common Polyforms, this is an equilateral triangle, a square or a regular hexagon. A Heptiamond contains seven cells in the form of equilateral triangles. This particular cell is the constituent element of the isometric grid, a type of ‘graph paper’ where the lines cross at sixty-degree angles.

The three
Clover Leaves

Clover Leaf
A particular two-piece Sub-Shape frequently encountered in solutions. It can be seen as two Jewels combined. There are three ways to construct one. The maximum number of Clover Leaves within a single solution is three.

Dudeney, Henry Ernest (1857-1930)
Famous English inventor of puzzles. He was the first to propose a Polyform problem, namely in his book The Canterbury Puzzles (1907), in this case, fitting Pentominoes into a Shape. The term Pentominoes, however, would be introduced later, by Golomb.

Family of three identical member Shapes

Family
A special type of Isomorphic Subdivision — an array of solutions in which a basic ‘Parent’ Shape is split up into Sub-Shapes that share their main characteristics with it, including those of symmetry (except in some ‘bastard’ cases). The example shows a Family with three member solutions of progressive subdivision. A Family is not a Metamorphosis in the proper sense, since each set of Sub-Shapes can be recombined into the original Shape.

The eleven Fists

Fist
A particular two-piece Sub-Shape frequently encountered in solutions. There are eleven ways to construct one. The maximum number of Fists within a single solution is four.

Flower and Miniflower

Flower
The most regular Shape to be solved with a set of Heptiamonds. It has True Hexagonal symmetry and its sides all measure two Basic Triangles. The Miniflower can be constructed with six Heptiamonds.

The six Gemstones

Gemstone
A particular two-piece Sub-Shape frequently encountered in solutions, bearing similarity to a Jewel. There are six ways to construct one. The maximum number of Fists within a single solution is four.

Golomb, Solomon Wolf (b. 1930)
American mathematician and engineer who coined the term Polyominoes in his Harvard lecture of 1953. His book Polyominoes — Puzzles, Patterns, Problems, and Packings (Princeton, 1965, revised and expanded 1994) is a standard for Polyforms enthousiasts. He often is mistakenly regarded as the inventor of the first Polyform problem; the credit for this goes to Dudeney.

Isometric grid

Isometric
Constructed of graphic elements that meet exclusively at sixty-degree angles and/or multiplications thereof. Heptiamonds are isometric.

Isomorphic Hole(s); example right
also has Isomorphic Subdivision

Isomorphic Hole(s)
A hole, or group of holes, in a Hollow Shape sharing the same form as the outer edges of a solution (but not necessarily its orientation).

Isomorphic Subdivision:
Equal Sizes (left) and Unequal Sizes (right)

Isomorphic Subdivision
A grouping of the Heptiamonds into a number of Sub-Shapes of the same form, but not necessarily of the same size. Therefore, a distinction is maintained between Equal and Unequal Sizes. The latter is found mainly in Section 7. Called Congruent Groups by Golomb.
See also Family.

Jewel: piece and
two Sub-Shapes

Jewel
The most compact of the Heptiamonds, bearing some similarity to the Gemstone In solutions they are often encountered in upscaled versions as Sub-Shapes, especially within the Flower Shape.

Three-stage Shape Metamorphosis

Metamorphosis
A gradual change in a Shape, presented through a series of solutions connected through arrows and a conclusive ‘period’.

• In a Shape Metamorphosis, the Shape itself changes its outer appearance in every stage. This includes Metamorphoses on holes and on Isomorphic Subdivision.
• In a Sub-Shape Metamorphosis, the Shape remains the same and only the puzzle pieces change positions.
• In an Internal Metamorphosis, a single set of Heptiamonds is divided over a group of progressively changing Sub-Shapes.

O'Beirne, Thomas H.
British matematician who dedicated much time in exploring Polyforms consisting of equilateral triangles. He named these Polyiamonds, after Golomb's term Polyominoes. The ending -iamond is borrowed from diamond, a rhombus containing two Basic Triangles. Like -ominoes, it is combined with Greek numerals into the names of animals, e.g. Triamond (having three BT) and ‘our’ Heptiamond.

Pentomino
Any of the twelve possible Polyforms that cover five squares laid side-to-side on a plane. These particular animals were incorporated in the earliest known Polyform puzzle problem, namely by Dudeney. The term itself was coined later by Golomb.

Three Polyforms: Pentomino,
Hexiamond, Pentahex

Polyform
The universally accepted term for a two-dimensional construct of equally shaped and sized polygons laid side to side on a plane. The most commonly found sets of Polyforms (animals) fall into three categories: Polyominoes (with squares), Polyiamonds (with equilateral triangles or Basic Triangles), and Polyhexes (with regular hexagons). Heptiamonds are within the Polyiamonds category. The term -iamond and its derivatives were introduced by O'Beirne.

Two Solid Shapes (left) and two Hollow Shapes (right)

Shape
The outlines of a solution, including separate ‘holes’ wherever present.

• A Solid Shape has no separate ‘holes’ inside its outer edge; the pieces fit tightly into a contiguous solution. Any Shape with apparent ‘holes’ touching the outer edge is still considered Solid (cp. second example).
• A Hollow Shape has at least one floating ‘hole’ fully disconnected from its outer edge.

Read the Introduction page for more information on Shapes in Heptiamonds.

Six and seven Steps

Steps Principle
Refers to those Shapes where the lay-out of Basic Triangles can be interpreted as a line of stacks that increase in height at regular intervals, the way stairs do. Steps do not necessarily coincide with Sub-Shapes.

Sub-Shape

Sub-Shape
A contiguous group of puzzle pieces of special interest; part of a Shape. The image shows a Flower Shape with a triangular Sub-Shape.

Torbijn, Pieter (d. 2007)
Dutch engineer and well-known inventor and solver of Polyforms problems. Torbijn made major contributions in the field of Heptiamonds. he solved his puzzles by hand, without the aid of a computer program (just like I do; must be in our common Dutch genes). Wherever a Heptiamond solution of his is shown on my site, it is quoted from this outside webpage.

Three Wieners

Wiener
One of the three elongated hexagonal Sub-Shapes with smooth edges, resulting from a specific Isomorphic Subdivision of a set of Heptiamonds. Related to the Bar.