Gemami Geometry

The inspiration for this article came from a geometry book I first discovered some 30 years ago on a dusty school cupboard. The book was published in 1890 and had a wealth of magnificent illustrations of practical geometric constructions. It was the chapter on construction of patterns, which contained a section on ornamental design- frets dating from Greek, Roman, Arabian, Moresque, Assyrians and Chinese designs, and a section on Arches and Geometrical Tracery Windows which I found inspirational. It is this last section

which I decided to explore during the lockdown in 2020.

Maybe you can challenge yourself to construct these shapes, or perhaps look around you and identify the types of arches used in any building you might happen upon.

All Constructions can be done using a pencil, ruler, a pair of compasses and a protractor (although in 1890, more use of set squares were suggested- who uses them now?)

The first construction is the semi-circular arch. This was used extensively by Romans to construct bridges, aqueducts and large-scale monuments (eg Colliseum). In most cases they did not use mortar, relying simply on the precision of their stone dressing. The wedge shaped blocks are called voussoirs.

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In the late Middle Ages, The Segmental arch was introduced, though the earliest example can be found in China from the 6th Century BC

This is an arch that has a rise that is less than a semi-circle. These flatter form of arches were commonly used for bridges, because larger spans were possible without excessively increasing the height.

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Construction Tip To find A, construct the perpendicular bisector of the span and experiment. If you want to fix the rise for a particular span, then construct the perpendicular bisector of a to b too.

When constructing segmental bridges, the ratio of rise to span had to be greater than 0.15.

This diagram shows a variation of the segmental arch, where the arch is made from arcs from 2 different points (Centres)

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The Arabs popularized the pointed arch. It was in their mosques that this form first acquired its religious connotations. Medieval Europe made great use of the pointed arch, which constituted a basic element in Gothic architecture. Pointed arches come in many forms, but the next three diagrams illustrate neatly the geometrical differences of an equilateral arch, an acute arch - known as a Lancet and an obtuse arch. All three arches are made from arcs of circles and A illustrates where to place the point of the pair of compasses. Note the radius of the equilateral arch must equal the span, in the acute arch the radius is more than the span and in the obtuse the radius is less than the span.

            Equilateral Arch                                              Acute Arch                                                         Obtuse Arch