Polygons and tilings, and sacred geometry

Laan pai CARTER \aaf e 5. Polygons and Tilings, and Sacred Geometry The Regular Polygons The Equilateral Triangle in, GENE Ry ow

11-10-11 10:45 Polygons, Tilings, & Sacred Geometry 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 Slide 5-1: Pompeii pavement 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. 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. Outline: Polygons EquilateralTriangle Tilings Hexagon & Hexagram Pentagon & Pentagram Golden Triangle Conclusion Reading Projects Polygons

11-10-11 10:45 Slide 5-23: 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). In this unit we'll see that each of these polygons also had symbolic meaning and appear in art motifs and architectural details, and some can be classified as sacred geometry. A polygon is a plane figure bounded by straight lines, called the sides of the polygon. From the Greek poly = many and gon = angle The sides intersect at points called the vertices. The angle between two sides is called an interior angle or vertex angle. Regular Polygons 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 Equilateral Triangle

11-10-11 10:45 Slide 5-2: Tablet in School of Athens, showing Tetractys Bouleau 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. Sacred Tetractys The Pythagoreans were particularly interested in this polygon because each triangular number forms an equilateral triangle. One special triangular number is the triangular number for what they called the decad, or ten, the sacred tetractys. Ten is important because it is, of course, the number of fingers. The tetractys became a symbol of the Pythagorean brotherhood. We've seen it before in the School of Athens. Trianglular Architectural Features

11-10-11 10:45 Slide 8-11: Church window in Quebec In architecture, triangular windows are common in churches, perhaps representing the trinity. Triskelion, Trefoil, Triquerta Other three-branched or three-comered designs include the triskelion. Slide 5-3: Greek Triskelion: Victory and Progress Lehner, Ernst. Symbols, Signs & Signets. NY: Dover, 1950 p. 85 Slide 5-4: Irish Triskelions from Book of Durrow. Met. Museum of Art. Treasures of Early Irish Art. NY: Met. 1977 Its a design that I liked so much I used it for one of my own pieces.

11-10-11 10:45 Slide 55: Calter carving Mandala II Calter photo Slide 56: Closeup of wheel Calter photo Tilings Slide 5-7: Pompeii Tiling with equilateral triangles Calter photo 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.

11-10-11 10:45 This is opposed to semiregular tilings like the Getty pavement shown here. Slide 5-8: Getty Pavement Calter photo The equilateral triangle is one of the three regular polygons that tile a plane. the other two being the square and hexagon. Hexagon & Hexagram Slide 5-15: Plate with Star of David Keller, Sharon. The Jews: A Treasury of Art and Literature. NY: Levin Assoc. 1992 Hexagonal Tilings Our next polygon is the hexagon, closely related to the equilateral triangle The hexagon is a favorite shape for tilings, as in these Islamic designs, which are not regular tilings, because they use more than one shape.

11-10-11 10:45 Slide 5-9: Islamic Tiling Patterns El-Said, Issam, et al. Geometric Concepts in Islamic Art. Palo Alto: Seymour, 1976. p. 54 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. Slide 510: Basket Calter photo

11-10-11 10:45 Slide 5-11: Pavement, Ducal Palace, Mantua Calter photo The Hexagon in Nature The hexagon is found in nature in the honeycomb, and some crystals such as basalt, and of course, in snowflakes. Slide 5-12: Snowflakes Bentley, W. A. Snow Crystals. NY: Dover, 1962. Six-Petalled Rose The hexagon is popular in architectural decoration partly because it is so easy to draw. In fact, these are rusty-compass constructions, which could have been made with a forked stick. Six circles will fit around a seventh, of the same diameter, dividing the circumference into 6 equal parts, and the radius of a circle exactly divides the circumference into six parts, giving a six petalled rose.

11-10-11 10:45 Slide 5-13: Moses Cupola. S. Marco, Venice Demus, Otto. The Mosaic Decoration of San Marco, Venice. Chicago: U. Chicago, 1988. plate 60. Hexagon vs. Hexagram Connecting alternate points of a hexagon gives a hexagram, a six-pointed star, usually called the Star of David, found in the flag of Israel. Slide 5-14: Star of David on Silver bowl from Damascus. Jewish Museum (New York, N.Y.), Treasures of the Jewish Museum. NY: Universe, 1986. p. 61 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.

11-10-11 10:45 Slide 5-17: The genii emerging. Burton, Richard. The Arabian nights entertainments. Ipswich : Limited Editions Club, 1954. The U.S. Great Seal Slide 520: Seal on Dollar Bill Calter photo 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

11-10-11 10:45 Quebec. Slide 5-22: Church Window in Quebec Calter photo This marvelous design is at Pompeii. It is made up of a central hexagon surrounded by squares, equilateral triangles, and rhombi. Slide: 5-23. Design at Pompeii Calter photo

11-10-11 10:45 Slide 5-24: Design on Pisa Duomo Calter photo This hexagram is one of countless designs on the Duomo in Pisa. Pentagon & Pentagram Slide 5-26: Pentagram from grave marker Calter photo 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.

11-10-11 10:45 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. Golden Triangle Slide 5-28: Emmer, plate F3 Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993. 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

11-10-11 10:45 Slide: 5-27: Penrose Tilings. Kappraff, Jay. Connections: The Geometric Bridge between Art & Science. NY: McGraw, 1990. p. 195 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. Slide 5-29: NCTM Cover Conclusion 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.

11-10-11 10:45 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! Reading Joseph Campbell, The Power of Myth, pp. 25-29 Carl Jung, Man andHis Symbols, pp. 266-285 Euclid, Elements, V2, pp. 97-104 Kappraff, Connections, pp. 85-87, 195-197 Fisher, p. 92-94 Projects Cut a circle from paper, fold in quarters vertically, then again horizontally, making a 4 x 4 grid. Mark the circumference where it crosses the grid. Connect these points in various ways to make the familiar regular polygons. 2 All these figures can be folded see Magnus Wenninger,Mathematics Through Paper Folding 4 Fold an equilateral triangle using NCTM method Construct a hexagon with compass 8 Construct a hexagon by paper folding, NCTM method 8 Construct a hexagon by folding a circle Make a pentagram by extending the sides of a pentagon, or make a pentagram by connecting the vertices of a pentagon Construct a pentagon by either of Euclid's methods. Connect the vertices to make a pentagram. 8 12 13 Construct a pentagon by the Medieval method. Connect the vertices 14 to make a pentagram Construct a pentagon by Durer's method 14 Check for in the pentagon by using dividers Solve the five-disk problem, Huntley p. 45 14 Put one type I and two type 11 golden triangles together to form a pentagon 14 15 Construct a triangle by Euclid's method 16 Construct a kite and a dart. Make xerox copies. Use them to make a 17 Penrose tiling. | <- Prev | Next - > | ©Paul Calter, 1998. All Rights Reserved. Dartmouth College.