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The axonemes of Gymnosphaera albida radiate from a central axoplast. Their proximal ends form a shell around the axoplast. Within the shell each axoneme is enveloped by a fibrillar sheath and the microtubules are interconnected by electron-dense linkages, which sometimes appear to be double. In nearly transverse sections the microtubules and their linkages form hexagons of 2 irregular types arranged in alternating rows. The shapes of the hexagons vary from one axoneme to the next. The variation is caused largely by the inclination of the axonemes to the line of sight, but also by distortion occurring during the preparation, observation and photography of the sections. Calculations show that, of a number of likely basic patterns (as would be seen in strictly transverse section), only one is compatible with measurements made on 9 of the axonemes. This involves only one type of hexagon oriented in 2 directions to form a ‘parquet-floor’ pattern. The hexagon is bilaterally symmetrical and its 6 microtubules all have the same set of angles between their linkages, namely an unpaired angle of 138 degrees 28′ and paired angles of 110 degrees 46′. Because these angles are in the ratio of 5:4:4, it is deduced that the microtubules have 13 protofilaments forming their walls. Morphogenetically the lateral growth of the pattern is governed by 2 rules: (1) there must be one, and only one, direction of 2-step zig-zagging of the linkages, and (2) linkages forming opposite sides of a hexagon must be in parallel.