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Drawn Mathematics

Mathematics you can see, draw, and fold.

Category 01 of 05

Ornamental Symmetry

Wallpaper groups, kaleidoscopes and star patterns drawn from their symmetry rules; examples are described by geometry, and any cultural origin is cited.

3 entries

  • Eight square panels of printed ornament in two rows: rosette grids, a basket weave, scrolls, nested squares, stepped diamonds, a six-petal lattice and one lone sunburst.

    Ornamental Symmetry

    Wallpaper Groups: Only 17 Symmetry Types for Two-Way Repeats

    Any flat pattern that repeats in two independent directions has one of 17 symmetry types. See all 17 drawn from one F, plate examples and a naming flowchart.

    Image credit

    "Nineveh & Persia No 2, Plate XIII, images 9 to 16 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551113).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped to the third and fourth rows of the plate (images 9 to 16 with their printed numbers); colours not altered

  • Olive bands edged in blue form eight-point stars around gold rosettes, with long red panels of leaf scrolls between them.

    Ornamental Symmetry

    Islamic Geometric Patterns: The 8-Point Star Is Two Squares

    Draw an eight-point star from one circle with a compass and straightedge, repeat it every 4 grid cells, and see why the pattern is p4m, or p4 once woven.

    Image credit

    "Moresque No 4, Plate XLII, image 2 (upper panel)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551279).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped to the upper panel of the plate (printed image 2), without the heading, the printed number or the lower panel; colours not altered

  • Illustration of a short paper-and-brass kaleidoscope in green-dotted paper, lying on its side on a green drafting table with its eyepiece end turned slightly towards the viewer and a pencil below it.

    Ornamental Symmetry

    Kaleidoscope: Lopsided Drawings Close Up at 60°, Not at 50°

    How a kaleidoscope works: a lopsided drawing repeats with no seam only if the mirror angle is 180° divided by a whole number: 60° works, while 50° and 72° fail.

    Image credit

    Illustration: Drawn Mathematics. Generated scene illustration — not a photograph, historical image or researched reconstruction. Use terms: may be reused with credit to "Drawn Mathematics"; provided without any claim of historical accuracy.