Review Vocabulary

Crystallographic notation A four-symbol notation used by crystallographers (and mathematicians) to classify strip patterns and wallpaper patterns. (p. 798)

Divine proportion Another glorifying term for the golden ratio. (p. 783)

Fibonacci numbers The numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, …. Each number after the second one is obtained by adding the two preceding numbers. (p. 780)

Fractal A pattern that exhibits similarity at ever-finer scales. (pp. p. 807 p. 808)

Generated A group is generated by a particular set of elements if composing them and their inverses in combinations can produce all elements of the group. (p. 806)

Geometric mean The geometric mean of two numbers and is (pp. p. 784 p. 785)

Glide reflection A glide reflection is a translation (= glide) combined with alternating reflection in a line parallel to the translation direction. Example: pbpbpb. (p. 792)

Golden ratio, golden mean Inflated names for the number (p. 783)

Golden rectangle A rectangle, the lengths of whose sides are in the golden ratio. (p. 783)

Group A group is a collection of elements with an operation on pairs of them such that the collection is closed under the operation, there is an identity for the operation, each element has an inverse, and the operation is associative. (p. 803)

Isometry Another word for rigid motion. Angles and distances, and consequently shape and size, remain unchanged by a rigid motion. For plane figures, there are only four possible isometries: reflection, rotation, translation, and glide reflection. (p. 791)

Phyllotaxis The spiral pattern of shoots, leaves, or seeds around the stem of a plant. (p. 782)

Preserves the pattern A transformation preserves a pattern if all parts of the pattern look exactly the same after the transformation has been performed. (p. 792)

Recursion A method of defining a sequence of numbers, in which the next number is given in terms of previous ones. (p. 781)

Reflection symmetry Mirror-image symmetry. (p. 792)

Rigid motion A transformation of the plane that preserves the size and shape of figures. In particular, any pair of points is the same distance apart after the transformation as before. (Also called isometry.) (p. 791)

Rosette pattern A pattern whose only symmetries are rotations about a single point and reflections through that point. (p. 793)

Rotational symmetry A figure has rotational symmetry if a rotation about its “center” leaves it looking the same, as occurs with the letter . (p. 796)

Strip pattern A pattern that has indefinitely many repetitions in one direction. (p. 793)

Symmetry of the pattern A transformation of a pattern is a symmetry of the pattern if it preserves the pattern. (p. 793)

Symmetry group of the pattern The group of symmetries that preserve the pattern. (p. 804)

Translation A rigid motion that moves everything a certain distance in one direction. (p. 791)

Translation symmetry An infinite figure has translation symmetry if it can be translated (slid, without turning) along itself without appearing to have changed. Example: . (p. 794)

Wallpaper pattern A pattern in the plane that has indefinitely many repetitions in more than one direction. (p. 793)