Finite Mathematics 1 Lesson 2: Tiling, Tessellations & Frieze Patterns

This flashcard set covers the vocabulary and concepts of tiling, tessellations, and the seven Frieze groups as presented in the Finite Mathematics 1 lecture.

Tiling (Tessellation)

A complete covering of a surface using one or more repeated, non-overlapping geometric shapes called tiles.

Regular Tessellations

Tessellations where tiles are made up of only one type of regular polygon; the only three polygons possible are equilateral triangles, squares, and regular hexagons.

Semi-regular (Archimedean) Tessellation

A tiling that combines two or more regular polygons such that the pattern at every vertex is the same; there are exactly $8$ semi-regular tessellations.

Monohedral Tessellation

A tessellation where all tiles are congruent, meaning it uses only one type of tile.

Periodic Tessellation

Tessellations in which the pattern repeats in a predictable manner, such as regular and semi-regular tessellations.

Aperiodic Tessellation

Tessellations with no repeating pattern.

Penrose Tilings

Aperiodic tilings formulated by mathematician Roger Penrose in the $1970$'s that use specific rules to avoid repeating patterns.

Irregular Tessellation

Tessellations comprised of non-regular, non-convex shapes.

Escher-type Tessellations

Tessellations or near-tessellations using recognizable, interlocking figures such as birds, lizards, or fish, popularized by M.C. Escher.

Frieze

The wide central section of molding and bands above the columns of a building in classical architecture.

The Hop (T)

A frieze pattern generated merely by translating a tile.

The Step (TG)

Frieze patterns generated by a glide reflection followed by a translation to ensure the pattern repeats infinitely.

The Sidle (TV)

A frieze pattern produced by a reflection of a tile with respect to a vertical line followed by a translation.

The Spinning Hop (TR)

A frieze pattern requiring a $180^\text{o}$ rotation of the tile followed by a translation.

The Spinning Sidle (TRVG)

A frieze pattern generated by a rotation or vertical reflection followed by a glide reflection and completed by translations.

The Jump (THG)

A frieze pattern group characterized by the use of horizontal reflections and then translations.

The Spinning Jump (TRHVG)

The most complex frieze group, requiring horizontal and vertical reflections followed by translation; the two reflections together are equivalent to a $180^\text{o}$ rotation.