Polygons and Polyhedrons

• Definitions
• Regular Polygons
• Regular 2-D Vertices
• Irregular 2-D Vertices
• Polygon Grids / Lattices
• 2-D Symmetry Types
• Regular 3-D Vertices
• Polygon Solids
• Polyhedron Grids / Lattices

Definitions

A polygon is a many-sided 2-dimensional closed shape. All the sides (= edges) must be straight lines. All the angles between edges must be greater than zero (and shouldn't be 180 degrees). A vertex is another name for a corner where the edges of polygons meet. There must be at least 3 polygons at each vertex to make a grid or a solid. If only 2 polygons are involved then they are sharing a bent edge rather than meeting at a true vertex.

A polyhedron is a many-sided 3-dimensional closed shape. All the sides (= faces) must be polygons. Note that the plural term may be polyhedrons or polyhedra.

The lower limit on the number of edges for a polygon is 3. It is impossible to construct a shape that holds area with less than 3 lines. A polygon with 3 edges is called a triangle (because of the three angles between them). A polygon with 4 edges is called a quadrilateral rather than a quadrangle! A polygon with 5 edges is called a pentagon, one with 6 edges is a hexagon and so on (using greek numbering). There is no upper limit on the number of edges but the polygon becomes hard to draw.

The lower limit on the number of faces for a polyhedron is 4. It is impossible to construct a shape that holds volume with less than 4 planes. There is no upper limit on the number of faces but the polyhedron becomes hard to construct.

A convex polygon is one with only interior angles less than 180 degrees. For example, a rectangle is convex but a chevron is not. The centre of gravity of a convex polygon is always within its area. The centre of gravity of a non-convex polygon may be inside or outside its area.

A regular polygon has all sides and all angles the same and therefore must also be convex. The regular 3-sided polygon is called an equilateral triangle. The 4-sided one is a square. There are no special names for the regular polygons with more sides. Simply specify a regular pentagon or hexagon etc.

There are special names for some irregular or semi-regular polygons though. An isosceles triangle has 2 of 3 angles the same and therefore 2 of 3 sides the same. A parallelogram has 4 sides and 4 angles and the opposite ones of each are the same. A rectangle is a parallelogram with all 4 angles the same - ie 90 degrees. Note that both diagonals of a rectangle are the same length but do not cross at 90 degrees. A rhombus is a parallelogram with all 4 sides the same. Note that the diagonals of a rhombus cross at 90 degrees but are not the same length. One special rhombus has one diagonal the same length as its sides but unfortunately has no special name.

A pentangle is a special non-convex polygon. It has 10 equal sides but 2 different angles (36 and 252 degrees) alternating to form a 5-pointed star. This shape is significant because it occurs in several non-convex polyhedrons. These are "stellations" of the regular convex ones.

A central angle is one between the lines (radii) joining 2 adjacent vertices to the centre. With a regular polygon, all central angles are the same. An interior angle is the angle between sides measured inside a vertex. With a regular polygon, all interior angles are the same. An exterior angle is the angle between sides measured outside a vertex when one of the sides is extended. With a regular polygon, all exterior angles are the same as the central angles. The central or exterior angle is 360 degrees divided by the number of sides. The interior angle is 180 degrees minus the central angle.

Regular Polygons

There are an infinite number of regular polygons (from 3 sides upwards). However, only a few can be used in a regular vertex.

Regular 2-D Vertices

The sum of the interior angles of the polygons at a 2-dimensional vertex must be equal to 360 degrees. The following table shows all possible permutations with regular polygons. Not all can be used in practice. Some vertex types can be repeated to fill the plane on their own. Some must be combined with others to fill the plane. Some are just not usable (ie triples with an odd-sided polygon and the others not being a pair).

Irregular 2-D Vertices

Here the error in the sum to 360 degrees is given to show how irregular the polygons must be. The smaller the error (negative or positive) the less irregular the polygons will look. Only convex polygons are considered here.

Polygon Grids / Lattices

Note that V stands for vertex, F for face, E for Edge and t for type. So Vt is vertex type and Vs is vertices in one unit cell of the grid. For my purposes, a unit cell is the smallest parallelogram that can be repeated in translations to make an infinite 2-dimensional grid. A crystallographer would choose a different (bigger) unit cell in some cases and would specify the symmetry a little differently.

There are only 3 regular and 8 semi-regular grids. Using vertex type as the primary characteristic, there are 18 demi-semi-regular grids. An objective measure of grid complexity might be the total number of edges in a unit cell divided by the motif repeats its symmetry requires. The lists of grids with arbitrary numbers of regular or irregular polygons can never be complete. However, note that for all grids: Vs + Fs = Es.

There are relationships between grids. Some are shown in the following table. Note:
V = original polygon grid (ie join original vertices).
truncate = cut off vertices (ie double polygon edges and leave space for a new polygon).
E = join centres of edges of polygon faces (ie expand / truncate vertices to halfway).
Er = join centres of adjacent polygons (ie the inverse grid from full truncation).
Vr = join vertices to centres of faces (ie expand / shave edges).
snub = reduce and twist polygons to leave space for more polygons (triangles).
* = some readjustment of edge length is required.

Note the square is its own inverse whereas the other grids are paired as inverses. As a result, its edges lead to a regular rhombus - squares again! Note that every rhombus is an inverse of a rectangle. The other special rhombus in the table above is the 60 degree one (like 2 equilateral triangles). It ought to have been given a special name too, eg a trombus (concatenating triangle and rhombus). The pentagon is irregular with two 90 degree and three 120 degree angles. The kite has two opposite 90 degree angles and a 60 degree angle opposite a 120 degree one.

2-D Symmetry Types

There are only 17 types of 2-dimensional symmetry. They are sometimes called plane groups. There are various standard nomenclatures for them but I use a minimalist classification system. The 2-D symmetry operations are rotation, mirror reflection and glide reflection. Translation should be reserved for repeating the unit cell to fill the plane but standard nomenclatures tend to include some. I disapprove of their "centred" cell as it arises from a prejudice that right angles are best.

Rotation symmetry repeats a motif by turning it about an axis. Only some rotations are possible in combination with translation to fill the plane. These are 2, 3, 4 and 6. A rotation of 2 is called a diad. The motif is turned round 180 degrees to repeat or match itself. A rotation of 3 is called a triad. The motif is turned round 120 degrees to repeat or match itself. A rotation of 4 is called a tetrad. The motif is turned round 90 degrees to repeat or match itself. A rotation of 6 is called a hexad. The motif is turned round 60 degrees to repeat or match itself. The presence of a triad inevitably leads to two more. The presence of a tetrad inevitably leads to a second tetrad and two diads. The presence of a hexad inevitably leads to triads and diads.

Mirror reflection should be obvious. The motif is flipped over to repeat or match itself. The mirror surface is a plane that intersects our plane along a line. Glide reflection is rather strange. The motif is flipped over to repeat itself half a unit cell away. The "glide" is this bit of translation before finishing the reflection.

The unit cell of a lattice is always some sort of parallelogram (opposite pairs of sides and angles being the same). Translation along its 2 axes repeats the motif to fill the plane. However, some parallelograms are special and are required by some symmetry groups. A rectangle has all 4 angles the same (right angles) and diagonals the same. This is required for mirrors parallel to cell walls. A rhombus has all 4 sides the same and diagonals crossing at right angles. This is required for mirrors diagonal to cell walls. The standard nomenclatures use the diagonals as cell walls, doubling the size of a unit cell to make a centred cell. So "p" is used for a primitive cell and "c" for a centred one. A centred cell has twice the number of motifs it should have. A square has all 4 sides & angles the same and diagonals the same & crossing at right angles. This is required for tetrads. A "trombus" (60/120 degree rhombus) has one diagonal the same length as its sides. This is required for triads and hexads. Bizarrely, the standard nomenclatures don't insist on a centred rectangular cell here!

In the following table of plane groups, I list both my classification and some standard ones. The symmetry defines the shape of the smallest possible unit cell and the number of motif repeats (M) within it. I also list the ratio of sides (R) and skew angle (S) used in my pattern software.

Symmetry occurs in nature but it is approximate rather than exact. Gravity has caused organisms to evolve with vertical mirror symmetry. As a result humans are prejudiced in favour of patterns that have vertical mirrors. Psychologically, vertical symmetry is restful while horizontal is restless. When no particular direction of travel is important, rotational symmetry occurs. For example: starfish and most flowers. Glide reflection combined with size change is seen in ferns. Rotation combined with size change or translation produces spirals and coils. For example: fir cones, pineapples and vines.

Islamic art tends to favour higher orders of rotation and as many mirror reflections as possible. This maximised the repetitiveness of their designs. They were only permitted to use abstract forms and the straight edges forced by mirrors were no problem. So most islamic designs are 6m or 4m. In contrast, M.C.Escher tended to avoid too many rotations and mirror reflections. They restrict the shapes in a design and Escher preferred recognisable animal and plant forms to abstract shapes. So most Escher designs are 1, 1g, 2 or 2g.

Regular 3-D Vertices

The sum of the interior angles of the polygons at a 3-dimensional polyhedron vertex must be less than 360 degrees. Not all possible permutations can be used on their own to make a solid.

Polygon Solids / Polyhedrons / Polyhedra

Note that V stands for vertex, F for face, E for Edge and t for type. So Vt is vertex type and Vs is vertices in the whole solid. Truncation refers to polygons of the inverse solid being added by shaving vertices. Shaving edges of regular polyhedrons adds polygons from the faces of rhombus solids. These can be stretched into regular squares or be broken into triangles to make twisted edge forms. Snubbing refers to the reduction and twisting of polygon faces, leaving space for these extra triangle faces.

There are only 5 regular solids (Platonic). There are 13 unique semi-regular solids (Archimedean) and 2 infinite series (prisms and twisted prisms). Using vertex type as the primary characteristic, the next simplest solids are the 2 bi-pyramids. The lists of solids with arbitrary numbers of regular or irregular polygons can never be complete. However, note that for all polyhedrons: Vs + Fs = Es + 2.

There are relationships between polyhedrons. Some are shown in the following table and its pictorial equivalent. Note:
V = original polyhedron (ie join original vertices).
truncate = cut off vertices (ie double polygon edges and leave space for a new polygon).
E = join centres of edges of polygon faces (ie shave vertices / truncate to halfway).
Er = join centres of adjacent polygons (ie the inverse solid from full truncation).
Vr = join vertices to external centres of faces (ie expand edges to planes)
* = some readjustment of edge length is required.

Note the tetrahedron is its own inverse whereas the other regular solids are paired as inverses. As a result, its edges lead to a regular rhombus solid - the squares of a cube! The halfway truncation of a tetrahedron is also regular - the octahedron. Performing the same edge operations on this as on the other inverse pairs is interesting. The icosahedron can be considered to be a snub/twisted octahedron, while the cuboctahedron is also a rhomb-octahedron.

In addition to convex solids there are also non-convex "stellations" of the regular polyhedra. These can be derived by more than one method. A list of so-called "uniform" polyhedra (convex and non-convex) has been proven complete. However this only includes those that meet certain restrictions. Some non-convex examples follow.

Polyhedron Grids / Lattices

A unit cell here is the smallest shape that can be repeated in translations to make an infinite 3-dimensional grid or lattice. A rhomboid is a 3-dimensional parallelogram (3 different edge lengths and 3 different angles between them). A cuboid is a 3-dimensional rectangle (3 different edge lengths but 90 degree angles between them).

Of the 5 regular polyhedrons, only 1 can make a regular grid on its own. This is the cube - 1 per unit cell! 8 cubes meet at each vertex (4 meet at an edge). The octahedron and tetrahedron together can make a semi-regular grid. 6 octahedrons and 8 tetrahedrons meet at each vertex (2 of each meet at an edge, alternating). The triangular prism can make a grid on its own. 12 prisms meet at each vertex (6 meet at each edge between squares and 4 meet at each triangle/square edge).