Crystallography & Symmetry
patterns derived from their groups
Concept

The Alhambra — where it appears

The palace in Granada whose tilework is the standard example of plane symmetry in ornament. Careful analysts have counted eleven, thirteen, fourteen and seventeen of the groups in it, and the disagreement is about what counts as an instance rather than about the mathematics.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

How often a dot gives the wrong group. Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it.

The groups ornament actually uses

Seventeen exist and decoration does not use them evenly. Which are common is an empirical question that published surveys answer differently, and part of the reason is a hazard this site can measure exactly.

classification · Ornament How often a dot gives the wrong group. Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it.

The Alhambra question

Textbooks say the Alhambra contains all seventeen wallpaper groups. Careful analysts of the same building have counted eleven, thirteen, fourteen and seventeen — and the disagreement is not about the mathematics but about what counts as an instance.

classification · Ornament

Named alongside it

The objects these essays reach for when they reach for this one.

Accidental symmetryGroup frequencyOrnamentPattern analysisColour symmetryTolerance

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