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Frieze Groups Tara Mccabe Denis Mortell

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Frieze Groups Tara Mccabe Denis Mortell Frieze Groups Tara McCabe Denis Mortell Introduction Our chosen topic is frieze groups. A frieze group is an infinite discrete group of symmetries, i.e. the set of the geometric transformations that are composed of rigid motions and reflections that preserve the pattern. A frieze pattern is a two-dimensional design that repeats in one direction. A study on these groups was published in 1969 by Coxeter and Conway. Although they had been studied by Gauss prior to this, Gauss did not publish his studies. Instead, they were found in his private notes years after his death. Visual Perspective Numerical Perspective What second-row numbers validate a frieze pattern? (F) There are seven types of groups. Each of these are the A frieze pattern is an array of natural numbers, displayed in a symmetry group of a frieze pattern. They have been lattice such that the top and bottom lines are composed only of The second-row numbers that determine the pattern are not illustrated below using the Conway notation and 1’s and for each unit diamond, random. Instead, these numbers have a peculiar property. corresponding diagrams: the rule (bc) − (ad) = 1 holds. Theorem I Fifth being a translation I The first, and the one There is a bijection between the valid which must occur in with vertical reflection, Here is an example of a frieze pattern: sequences for frieze patterns and all others is glide reflection and 180◦ rotation, or a the number of triangles adjacent to the Translation, or a vertices of a triangulated polygon. [2] ‘hop’. ‘spinning slide’. The triangulated polygon on the right The second is a glide determines the lattice example from I I Sixth being translation [2] earlier. It has 7 sides, which indicate the with translation, or a with horizontal glide period of the frieze pattern. Note that the ‘step’. reflections, or a ‘jump’. adjacent triangles to each vertex matches Properties the second-row numbers, starting at . I The third is a translation with a I The last is a horizontal I Every pattern of order n (i.e. with n lines) admits an Applications vertical reflection, or and vertical reflection horizontal translation of length n + 1. with translation 180◦, or a ‘slide’. I Every pattern of order n is completely determined by a Frieze patterns are used very often in decorative art and a ’spinning jump’. sequence of n + 1 numbers. architecture worldwide and for centuries, before frieze group I Every pattern is completely determined by the numbers on theory even surfaced. Friezes in architecture are long stretches of the second line. ( ) I The fourth is a F painted, sculpted or calligraphic decorations, sometimes depicting translation with a scenes in a sequence of discrete panels. They are used most 180◦ rotation, or a I The pattern formed famously in Greek and Roman architecture. ‘spinning hop’. above using is a frieze pattern! It is a References Frieze groups have been used to develop an algorithm to detect translation with a 180◦ repeating patterns in images. This can be used for object Y. Liu and R.T. Collins. “Frieze and Wallpaper Symmetry Groups Classification under Affine and Perspective Distortion”. In: rotation. (1999). recognition in images, even where the pattern is distorted by J.F. Marceau. “Frieze Patterns and Triangulated Polygons”. In: (2013). perspective [1]..
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