Inscribed figures

Tile Work of the Lower walls of Salon del Trono, Alhambra, Spain
Field: Geometry
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Tile Work of the Lower walls of Salon del Trono, Alhambra, Spain

Detailed here is the lower wall of a throne room in the Spanish Muslim palace of the Alhambra. It is considered one of the finest examples of ceramic inlay, radiative geometry, and use of inscribed figures in art.

Basic Description

An inscribed or "cyclic" figure is one which fits "snugly" into another larger shape; every vertex of the enclosed figure must touch the edge of the enclosing figure. An inscribed quadrilateral will always follow Ptolemy's Theorem, and all regular polygons are cyclic.

applications

Inscribed squares are, amongst other things, a proof of the pythagorean theorem. Certain figures are noticeably easier to construct if thought of as an inscription of another figure, notably stars. (construction of a star)

A More Mathematical Explanation

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Inscribed Figures applications

===Example 1: constructing a star and The Walls of The Salon d [...]

[Click to hide A More Mathematical Explanation]

Inscribed Figures applications

Example 1: constructing a star and The Walls of The Salon del Trono

The design of above is quite complicated to the point that it would hard to imagine how it could be easily created. if the design is thought of in terms of a series of inscribed figures based on a singular inscribed baseline figure(the central star), designing so complex a pattern is conceivable.The design detailed above is generated from the twelve pointed star at the center, which in turn is designed from a series of three inscribed squares.

Each point is spaced a quarter of the length of the original squares’ side.At every place where the inscriptions’ extended edges meet the edges of the original rectangle, a point is made. the meeting of the first inscription with the orig gives halfway marks, while the extended edges of the innermost inscribed square intersect the edges of the outermost square at the quarter lengths of original square. The points of intersection between extended edges of inscriptions and the edges of the original square are shown in the diagram

The rays connecting every quarter mark to the nearest perpendicular halfway mark and vice versa will generate a twelve pointed star.

Connecting the edges of the star with every other edge yields the outermost points of the central star.

If these schematics are overlaid with the original star, the congruence of construction becomes apparent:

Example 2:Erecting the London Eye

To construct the London eye, the engineers thought of the circle as series of isoceles triangle sections, which together form an inscribed pentagon. A circle is harder to fabricate and accurately recreate at any scale owing to the sheer number of elements it possesses(the infinite points which make up the circumference). But a a cyclic polygon which can be divided into smaller elements is still easier to work with. to efficiently put the wheel together, broke up into arcs, which were based around the easier-to-construct inscribed polygon:

See the pentagon, then see the smaller sections and then where the circle fits into the edges, and one can see how the circle can be easily constructed:

Teaching Materials

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References

Remarkable Structures: Engineering Today's Innovative Buildings. Sutherland 2002

Future Directions for this Page

Fixed References, elaboration on the london eye

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