A heptomino (or 7-omino) 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 pop ...

of order 7, that is, a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

in the

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

made of 7 equal-sized

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

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

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

s are not considered to be distinct shapes, there are

108 108 may refer to: * 108 (number) * AD 108, a year * 108 BC, a year * 108 (artist) (born 1978), Italian street artist * 108 (band), an American hardcore band * 108 (emergency telephone number), an emergency telephone number in several states in Ind ...

different ''free'' heptominoes. When reflections are considered distinct, there are

196 Year 196 ( CXCVI) was a leap year starting on Thursday (link will display the full calendar) 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 ...

''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 groups: *84 heptominoes (coloured grey) have no symmetry. 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. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D the ...

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 has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...

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 of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...

of order 2, also known as the

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

. *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 sufficient rule for when a prototile will tile the plane. It consists of the following requirements:Will It Tile? Try ...

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, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or o ...

a rectangle 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 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"
fro

/ref>

References

{{Polyforms Polyforms