Description and Requirements The Book Bibliography Syllabus Introduction The Great Pyramid Music of the Spheres Number Symbolism Polygons and Tilings The Platonic Solids Roman Architecture Number Symbolism in the Middle Ages The Wheel of Fortune Celestial Themes in Art Origins of Perspective What Shape Frame? Piero della Francesca Leonardo Façade measurement by Trigonometry Early Twentieth Century Art Dynamic symmetry & The Spiral The Geometric Art of M.C. Escher Later Twentieth Century Geometry Art Art and the Computer Chaos & Fractals

Polygons, Tilings,
Slide 51: Pompeii pavement
In the last unit, Number Symbolism, we saw that in the ancient world certain numbers had symbolic meaning, aside from their ordinary use for counting or calculating. In this unit we'll show that the plane figures, the polygons, triangles, squares, hexagons, and so forth, were related to the numbers (three and the triangle, for example), were thought of in a similar way, and in fact, carried even more emotional baggage than the numbers themselves, because they were visual. This takes us into the realm of Sacred Geometry. For now we'll do the polygons directly related to the Pythagoreans; the equilateral triangle (Sacred tetractys), hexagon, triangular numbers, and pentagram. We'll also introduce tilings, the art of covering a plane surface with polygons.
Slide 523: Design at Pompeii Calter photo
In the last unit, Number Symbolism we saw that in the ancient
world certain numbers had symbolic meaning, aside from their ordinary
use for counting or calculating. But each number can be associated with
a plane figure, or polygon (Three and the Triangle, for
example). A regular polygon is one in which all the sides and interior angles are equal. Polygons vs. Polygrams A polygram can be drawn by connecting the vertices of a polgon. Pentagon & Pentagram, hexagon & hexagram, octagon & octograms
There are, of course, an infinite number of regular polygons, but we'll
just discuss those with sides from three to eight. In this unit we'll
cover just those with 3, 5, and 6 sides. We'll start with the
simplest of all regular polygons, the equilateral triangle. Slide 811: Church window in Quebec In architecture, triangular windows are common in churches, perhaps representing the trinity. Triskelion, Trefoil, TriquertaOther threebranched or threecomered designs include the triskelion.
Slide 53: Greek Triskelion: Victory and Progress
Slide 54: Irish Triskelions from Book of Durrow.
Tilings or tesselations refers to the complete covering of a plane surface by tiles. There are all sorts of tilings, some of which we'll cover later. For now, lets do the simplest kind, called a regular tiling, that is, tiling with regular polygons. This is opposed to semiregular tilings like the Getty pavement shown here.
The equilateral triangle is one of the three regular polygons that tile a plane. the other two being the square and hexagon.
Hexagonal Tilings
Our next polygon is the hexagon, closely related to the equilateral triangle Slide 59: Islamic Tiling Patterns
But, as we saw, the hexagon is one of the three regular polygons will make a regular tiling. An Illusion The hexagon is sometimes used to create the illusion of a cube by connecting every other vertex to the center, forming three diamonds, and shading each diamond differently.
The Hexagon in Nature The hexagon is found in nature in the honeycomb, and some crystals such as basalt, and of course, in snowflakes.
SixPetalled Rose
The hexagon is popular in architectural decoration partly because it is
so easy to draw. In fact, these are rustycompass constructions,
which could have been made with a forked stick.
Hexagon vs. Hexagram Connecting alternate points of a hexagon gives a hexagram, a sixpointed star, usually called the Star of David, found in the flag of Israel.
Slide 514: Star of David on Silver bowl from Damascus.
Solomon's Seal The hexagrarn is also called a Solomon's Seal. Joseph Campbell says that King Solomon used this seal to imprison monsters & giants into jars.
The U.S. Great Seal
The hexagrarn can also be viewed as two overlapping Pythagorean tetractys. Joseph Campbell writes; In the Great Seal of the U.S. there are two of these interlocking triangles. We have thirteen points, for our original thirteen states, and six apexes: one above, one below, andfour to thefour quarters. The sense of this might be thalftom above or below, orftom any point of the compass, the creative word may be heard, which is the great thesis of democracy.  The Power of Myth. p.27 Hexagonal Designs in Architecture Hexagonal designs are common in ancient architecture, such as this church window in Quebec.
This marvelous design is at Pompeii. It is made up of a central hexagon surrounded by squares, equilateral triangles, and rhombi.
Slide: 523. Design at Pompeii
Slide 524: Design on Pisa Duomo
This hexagram is one of countless designs on the Duomo in Pisa.
The Pentagram was used as used as a sign of salutaton by the Pythagoreans, its construction supposed to have been a jealously guarded secret. Hippocrates of Chios is reported to have been kicked out of the group for having divulged the construction of the pentagram. The pentagram is also called the Pentalpha, for it can be thought of as constructed of five A's. Euclid's Constructions of the Pentagon Euclid gives two constructions in Book IV, as Propositions 11 & 12. According to the translator T.L. Heath, these methods were probably developed by the Pythagoreans. Medieval Method of Construction Supposedly this construction was one of the secrets of Medieval Mason's guilds. It can be found in Bouleau p. 64. Durer's Construction of the Pentagon Another method of construction is given in Duret's "Instruction in the Measurement with the Compass and Ruler of Lines, Surfaces and Solids," 1525. Its the same construction as given in Geometria Deutsch, a German book of applied geometry for stonemasons and Golden Ratios in the Pentagram and Pentagon The pentagon and pentagram are also interesting because they are loaded with Golden ratios, as shown in Boles p.48.
Slide 528: Emmer, plate F3
The Golden Triangle A golden triangle also called the sublime triangle, is an isoceles triangle whose ratio of leg to base is the golden ratio. It is also an isoceles triangle whose ratio of base to leg is the golden ratio, so there are two types: Type I, acute, and type II, obtuse. A pentagon can be subdivided into two obtuse and one acute golden triangle. Euclid's Construction Euclid shows how to construct a golden triangle. Book IV, Proposition 10 states, "To construct an isoceles triangle having each of the angles at the base the double of the remaining one." Penrose Tilings
One place that the golden triangle appears is in the Penrose Tiling, invented by Roger Penrose, in the late seventies. The curious thing about these tilings is they use only two kinds of tiles, and will tile a plane without repeating the pattern. Making a Penrose Tiling A Penrose tiling is made of two kinds of tiles, called kites and darts. A kite is made from two acute golden triangles and a dart from two obtuse golden triangles, as shown above.
So we covered the triangle, pentagon, and hexagon, with sides 3, 5, and 6. We'll cover the square and octagon in a later unit. Its clear that these figures, being visual, carried even more powerful emotional baggage than the numbers they represent. Next time we'll again talk about polygons, in particular the triangle. But I won't waste your time with some insignificant and trivial fact about the triangle, but will show that, according to Plato, triangles form the basic building block of the entire universe!
Joseph Campbell, The Power of Myth, pp. 2529Projects
©Paul Calter, 1998. All Rights Reserved. Dartmouth College.
