A heptomino (or 7-omino or septomino) is a

polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popu ...

of order 7; that is, a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

in the plane made of 7 equal-sized

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s connected edge to edge. The name of this type of figure is formed with the prefix hept(a)-. When

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

s and reflections are not considered to be distinct shapes, there are 108 different ''free'' heptominoes. When reflections are considered distinct, there are

196 Year 196 ( CXCVI) was a leap year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Dexter and Messalla (or, less frequently, year 949 ''Ab urbe condita''). The denomination 196 for this yea ...

''one-sided'' heptominoes. When rotations are also considered distinct, there are 760 ''fixed'' heptominoes.

Symmetry

The figure shows all possible free heptominoes, coloured according to their

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s: *84 heptominoes (coloured grey) have no

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

. Their symmetry group consists only of the

identity mapping Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

. *9 heptominoes (coloured red) have an axis of

reflection symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...

aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares. :: *7 heptominoes (coloured green) have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection. :: *4 heptominoes (coloured blue) have point symmetry, also known as

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...

of order 2. Their symmetry group has two elements, the identity and the 180° rotation. :: *3 heptominoes (coloured purple) have two axes of reflection symmetry, both aligned with the gridlines. Their symmetry group has four elements, the identity, two reflections and the 180° rotation. It is the

dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

of order 2, also known as the

Klein four-group In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...

. *1 heptomino (coloured orange) has two axes of reflection symmetry, both aligned with the diagonals. Its symmetry group also has four elements. Its symmetry group is also the dihedral group of order 2 with four elements. :: If reflections of a heptomino are considered distinct, as they are with one-sided heptominoes, then the first and fourth categories above would each double in size, resulting in an extra 88 heptominoes for a total of 196. If rotations are also considered distinct, then the heptominoes from the first category count eightfold, the ones from the next three categories count fourfold, and the ones from the last two categories count twice. This results in 84 × 8 + (9+7+4) × 4 + (3+1) × 2 = 760 fixed heptominoes.

Packing and tiling

Of the 108 free heptominoes, 101 satisfy the

Conway criterion In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a Necessity and sufficiency, sufficient rule for when a prototile will tile the plane. It consists of the following req ...

and 3 more can form a patch satisfying the criterion. Thus, only 4 heptominoes fail to satisfy the criterion and, in fact, these 4 are unable to tessellate the plane. : Although a complete set of the 108 free heptominoes has a total of 756 squares, it is not possible to

tile Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, wal ...

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

with that set. The proof of this is trivial, since there is one heptomino which has a hole. It is also impossible to pack them into a 757-square rectangle with a one-square hole because 757 is a prime number. However, the set of 107

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...

free heptominoes—that is, the ones without the hole—can tile a 7 by 107 (749-square) rectangle. Furthermore, the complete set of free heptominoes can tile three 11-by-23 (253-square) rectangles, each with a one-square hole in the center; the complete set can also tile twelve 8 by 8 (64-square) squares with a one-square hole in the "center".Image, "An incredible heptomino solution by Patrick Hamlyn"
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References

{{Polyforms Polyforms