Sacred Geometry Introductory Tutorial by Bruce Rawles

8/10/2019 Sacred Geometry Introductory Tutorial by Bruce Rawles

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Sacred Geometry Introductory Tutorial by Bruce Rawles

In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life

discernible by human eyes, to the greater cosmos. These inevitably follow geometrical archetypes, which reveal to us

the nature of each form and its vibrational resonances. They are also symbolic of the underlying metaphysical

principle of the inseparable relationship of the part to the whole. It is this principle of oneness underlying all geometry

that permeates the architecture of all form in its myriad diversity. This principle of interconnectedness, inseparability

and union provides us with a continuous reminder of our relationship to the whole, a blueprint for the mind to the

sacred foundation of all things created.

The Sphere

(charcoal sketch of a sphere byNancy Bolton-Rawles)

Starting with what may be the simplest and most perfect of forms, the sphere is an ultimate expression of unity,

completeness, and integrity. There is no point of view given greater or lesser importance, and all points on the

surface are equally accessible and regarded by the center from which all originate. Atoms, cells, seeds, planets, and

globular star systems all echo the spherical paradigm of total inclusion, acceptance, simultaneous potential and

fruition, the macrocosm and microcosm.

The Circle

The circle is a two-dimensional shadow of the sphere which is regarded throughout cultural history as an icon of the

ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects

of the profound and consummate perfection of the circle, sphere and other higher dimensional forms of these we

might imagine.

The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi

equals about 3.14159265358979323846264338327950288419716939937511) It cannot be expressed in terms of

the ratio of two whole numbers, or in the language of sacred symbolism, the essence of the circle exists in a

dimension that transcends the linear rationality that it contains. Our holistic perspectives, feelings and intuitions
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encompass the finite elements of the ideas that are within them, yet have a greater wisdom than can be expressed by

those ideas alone.

The Point

At the center of a circle or a sphere is always an infinitesimal point. The point needs no dimension, yet embraces all

dimension. Transcendence of the illusions of time and space result in the point of here and now, our most primal light

of consciousness. The proverbial light at the end of the tunnel is being validated by the ever-increasing literature on

so-called near-death experiences. If our essence is truly spiritual omnipresence, then perhaps the point of our

being here is to recognize the oneness we share, validating all individuals as equally precious and sacred aspects

of that one.

Life itself as we know it is inextricably interwoven with geometric forms, from the angles of atomic bonds in the

molecules of the amino acids, to the helical spirals of DNA, to the spherical prototype of the cell, to the first few cells

of an organism which assume vesical, tetrahedral, and star (double) tetrahedral forms prior to the diversification of

tissues for different physiological functions. Our human bodies on this planet all developed with a common geometricprogression from one to two to four to eight primal cells and beyond.

Almost everywhere we look, the mineral intelligence embodied within crystalline structures follows a geometry

unfaltering in its exactitude. The lattice patterns of crystals all express the principles of mathematical perfection and

repetition of a fundamental essence, each with a characteristic spectrum of resonances defined by the angles,

lengths and relational orientations of its atomic components.

The Square Root of Two

The square root of 2 embodies a profound principle of the whole being more than the sum of its parts. (The square

root of two equals about 1.414213562) The orthogonal dimensions (axes at right angles) form the conjugal union of

the horizontal and vertical which give birth to the greater offspring of the hypotenuse. The new generation possesses

the capacity for synthesis, growth, integration and reconciliation of polarities by spanning both perspectives equally.

The root of two originating from the square leads to a greater unity, a higher expression of its essential truth, faithful

to its lineage.

The fact that the root is irrational expresses the concept that our higher dimensional faculties cant always necessarily

be expressed in lower order dimensional termse.g. And the light shineth in darkness; and the darkness

comprehended it not. (from the Gospel of St. John, Chapter 1, verse 5). By the same token, we have the capacity to

surpass the genetically programmed limitations of our ancestors, if we can shift into a new frame of reference (i.e.

neutral with respect to prior axes, yet formed from that matrix-seed conjugation. Our dictionary refers to the word


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matrix both as a womb and an array (or grid lattice). Our language has some wonderful built-in metaphors if we look

for them!

The Golden Ratio

The golden ratio (a.k.a. phi ratio a.k.a. sacred cut a.k.a. golden mean a.k.a. divine proportion) is another fundamental

measure that seems to crop up almost everywhere, including crops. (The golden ratio is about

1.618033988749894848204586834365638117720309180) The golden ratio is the unique ratio such that the ratio

of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. As such, it

symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for

retracing its lineage.

The golden ratio (phi) has some unique properties and makes some interesting appearances:

phi = phi^21; therefore 1 + phi = phi^2; phi + phi^2 = phi^3; phi^2 + phi^3= phi^4; ad infinitum.

phi = (1 + square root(5)) / 2 from quadratic formula, 1 + phi = phi^2.


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phi = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/)))))

phi = 1 + square root(1 + square root(1 + square root(1 + square root(1 + square root(1 + )))))

phi = (sec 72)/2 =(csc 18)/2 = 1/(2 cos 72) = 1/(2 sin 18) = 2 sin 54 = 2 cos 36 = 2/(csc 54) = 2/ (sec 36) for all you

trigonometry enthusiasts.

phi = the ratio of segments in a 5-pointed star (pentagram) considered sacred to Plato and Pythagoras in their

mystery schools. Note that each larger (or smaller) section is related by the phi ratio, so that a power series of the

golden ratio raised to successively higher (or lower) powers is automatically generated: phi, phi^2, phi^3, phi^4,

phi^5, etc.

phi = apothem to bisected base ratio in theGreat Pyramid of Giza

phi = ratio of adjacent terms of the famousFibonacci Seriesevaluated at infinity; the Fibonacci Series is a rather

ubiquitous set of numbers that begins with one and one and each term thereafter is the sum of the prior two

terms, thus: 1,1,2,3,5,8,13,21,34,55,89,144 (interesting that the 12th term is 12 raised to a higher power,

which appears prominently in a vast collection of metaphysical literature)

The mathematician credited with the discovery of this series isLeonardo Pisano Fibonacciand there is a publication

devoted to disseminating information about its unique mathematical properties,The Fibonacci Quarterly

Fibonacci ratios appear in the ratio of the number of spiral arms in daisies, in the chronology of rabbit populations, in

the sequence of leaf patterns as they twist around a branch, and a myriad of places in nature where self-generating

patterns are in effect. The sequence is the rational progression towards the irrational number embodied in the

quintessential golden ratio.

This most aesthetically pleasing proportion, phi, has been utilized by numerous artists since (and probably before!)

the construction of the Great Pyramid. As scholars and artists of eras gone by discovered (such as Leonardo da

Vinci,Plato,andPythagoras), the intentional use of these natural proportions in art of various forms expands our

sense of beauty, balance and harmony to optimal effect. Leonardo da Vinci used the Golden Ratio in his painting

ofThe Last Supperin both the overall composition (three vertical Golden Rectangles, and a decagon (which containsthe golden ratio) for alignment of the central figure of Jesus.

The outline of theParthenonat the Acropolis near Athens, Greece is enclosed by a Golden Rectangle by design.

The Square Root of 3 and the Vesica Piscis
http://www.zazzle.com/the_sphinx_and_the_great_pyramid_at_giza_posters-228812889715461361http://www.zazzle.com/the_sphinx_and_the_great_pyramid_at_giza_posters-228812889715461361http://www.zazzle.com/the_sphinx_and_the_great_pyramid_at_giza_posters-228812889715461361http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.htmlhttp://www.fq.math.ca/http://www.fq.math.ca/http://www.fq.math.ca/http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Plato.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Plato.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Plato.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.htmlhttp://goldennumber.net/art.htmhttp://goldennumber.net/art.htmhttp://goldennumber.net/art.htmhttp://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#parthenonhttp://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#parthenonhttp://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#parthenonhttp://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#parthenonhttp://goldennumber.net/art.htmhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Plato.htmlhttp://www.fq.math.ca/http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.htmlhttp://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/http://www.zazzle.com/the_sphinx_and_the_great_pyramid_at_giza_posters-228812889715461361


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TheVesica Piscisis formed by the intersection of two circles or spheres whose centers exactly touch. This symbolic

intersection represents the common ground, shared vision or mutual understanding between equal individuals.

The shape of the human eye itself is a Vesica Piscis. The spiritual significance of seeing eye to eye to the mirror of

the soul was highly regarded by numerous Renaissance artists who used this form extensively in art and

architecture. The ratio of the axes of the form is the square root of 3, which alludes to the deepest nature of the triune

which cannot be adequately expressed by rational language alone.

Spirals

This spiral generated by a recursive nest of Golden Triangles (triangles with relative side lengths of 1, phi and phi) isthe classic shape of the Chambered Nautilus shell. The creature building this shell uses the same proportions for

each expanded chamber that is added; growth follows a law which is everywhere the same. The outer triangle is the

same as one of the five arms of the pentagonal graphic above.

Toroids
http://www.geometrycode.com/free/how-to-calculate-the-area-of-the-vesica-piscis/http://www.geometrycode.com/free/how-to-calculate-the-area-of-the-vesica-piscis/http://www.geometrycode.com/free/how-to-calculate-the-area-of-the-vesica-piscis/http://www.geometrycode.com/free/how-to-calculate-the-area-of-the-vesica-piscis/


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Rotating a circle about a line tangent to it creates a torus, which is similar to a donut shape where the center exactly

touches all the rotated circles. The surface of the torus can be covered with 7 distinct areas, all of which touch each

other; an example of the classic map problem where one tries to find a map where the least number of unique

colors are needed. In this 3-dimensional case, 7 colors are needed, meaning that the torus has a high degree of

communication across its surface. The image shown is a birds-eye view.

Dimensionality

The progression from point (0-dimensional) to line (1-dimensional) to plane (2-dimensional) to space (3-dimensional)

and beyond leads us to the questionif mapping from higher order dimensions to lower ones loses vital information

(as we can readily observe with optical illusions resulting from third to second dimensional mapping), does our

fixation with a 3-dimensional space introduce crucial distortions in our view of reality that a higher-dimensional

perspective would not lead us to?

Fractals and Recursive Geometries


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There is a wealth of good literature on this subject; its always fascinating how nature propagates the same essence

regardless of the magnitude of its expressionour spirit is spaceless yet can manifest aspects of its individuality at

any scale.

Perfect Right Triangles

The 3/4/5, 5/12/13 and 7/24/25 triangles are examples of right triangles whose sides are whole numbers. The graphic

above contains several of each of these triangles. The 3/4/5 triangle is contained within the so-called Kings


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Chamber of the Great Pyramid, along with the 2/3/root5 and 5/root5/2root5 triangles, utilizing the various diagonals

and sides.

The Platonic Solids

Here are LOTS of math details and images of the Platonic Solids and Archimedean

Solids.

The 5 Platonic solids (Tetrahedron, Cube or (Hexahedron), Octahedron, Dodecahedron and Icosahedron) are ideal,

primal models of crystal patterns that occur throughout the world of minerals in countless variations. These are the

only five regular polyhedra, that is, the only five solids made from the same equilateral, equiangular polygons. To the

Greeks, these solids symbolized fire, earth, air, spirit (or ether) and water respectively. The cube and octahedron are

duals, meaning that one can be created by connecting the midpoints of the faces of the other. The icosahedron and

dodecahedron are also duals of each other, and three mutually perpendicular, mutually bisecting golden rectangles

can be drawn connecting their vertices and midpoints, respectively. The tetrahedron is a dual to itself.

From April, 2000 through (at least) December, 2003, we conducted a poll on which Platonic Solid folks liked best;

heres the results from that period. If theres interest, Ill start another poll to see if the preferences has shifted!
http://www.geometrycode.com/free/polyhedra-illustrations-of-platonic-and-archimedean-solids/http://www.geometrycode.com/free/polyhedra-illustrations-of-platonic-and-archimedean-solids/http://www.geometrycode.com/free/polyhedra-illustrations-of-platonic-and-archimedean-solids/http://www.geometrycode.com/free/polyhedra-illustrations-of-platonic-and-archimedean-solids/http://www.geometrycode.com/free/polyhedra-illustrations-of-platonic-and-archimedean-solids/


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Here are someanimations of counter-rotating polyhedra and images of the Platonic solids showing their relationships

as duals.

Here arefold-up patterns for the Platonic Solids.

The Archimedean Solids
http://www.geometrycode.com/free/rotating-star-tetrahedra-polyhedra-animations-and-ray-traced-images/http://www.geometrycode.com/free/rotating-star-tetrahedra-polyhedra-animations-and-ray-traced-images/http://www.geometrycode.com/free/rotating-star-tetrahedra-polyhedra-animations-and-ray-traced-images/http://www.geometrycode.com/free/rotating-star-tetrahedra-polyhedra-animations-and-ray-traced-images/http://www.geometrycode.com/free/platonic-solids-fold-up-patterns/http://www.geometrycode.com/free/platonic-solids-fold-up-patterns/http://www.geometrycode.com/free/platonic-solids-fold-up-patterns/http://www.geometrycode.com/free/platonic-solids-fold-up-patterns/http://www.geometrycode.com/free/rotating-star-tetrahedra-polyhedra-animations-and-ray-traced-images/http://www.geometrycode.com/free/rotating-star-tetrahedra-polyhedra-animations-and-ray-traced-images/


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There are 13 Archimedean solids, each of which are composed of two or more different regular polygons.

Interestingly, 5 (Platonic) and 13 (Archimedean) are both Fibonacci numbers, and 5, 12 and 13 form a perfect right

triangle.

Here arefold-up patterns for the Archimedean Solids.

Stellations of The Platonic Solids and The Archimedean Solids
http://www.geometrycode.com/free/archimedean-solids-fold-up-patterns/http://www.geometrycode.com/free/archimedean-solids-fold-up-patterns/http://www.geometrycode.com/free/archimedean-solids-fold-up-patterns/http://www.geometrycode.com/free/archimedean-solids-fold-up-patterns/


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This is a stellation of a dodecahedron where each pentagonal face is capped with a pentagonal pyramid composed of

5 golden triangles, a sort of 3-dimensional 5-pointed star.

Here are more images of polyhedra (Platonic and Archimedean Solids.)

Metatrons Cube

Metatrons Cube contains 2-dimensional images of the Platonic Solids (as shown above) and many other primal

forms.

The Flower of Life
http://web.eecs.utk.edu/~plank/plank/origami/kaleido_pics/pics.htmlhttp://web.eecs.utk.edu/~plank/plank/origami/kaleido_pics/pics.htmlhttp://web.eecs.utk.edu/~plank/plank/origami/kaleido_pics/pics.html


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Indelibly etched on the walls of temple of the Osirion at Abydos, Egypt, the Flower of Life contains a vast Akashic

system of information, including templates for the five Platonic Solids. The background graphic for this page is a

repetitive hexagonal grid based on the Flower of Life. Below is an example ofFlower of Life jewelry.

For further information, visit myResources page.

Polygons, Tilings,&

Sacred Geometry
http://www.ka-gold-jewelry.com/p-articles/flower-of-life.phphttp://www.ka-gold-jewelry.com/p-articles/flower-of-life.phphttp://www.ka-gold-jewelry.com/p-articles/flower-of-life.phphttp://www.geometrycode.com/resources/http://www.geometrycode.com/resources/http://www.geometrycode.com/resources/http://www.ka-gold-jewelry.com/p-articles/flower-of-life.phphttp://www.ka-gold-jewelry.com/p-articles/flower-of-life.phphttp://www.geometrycode.com/resources/http://www.ka-gold-jewelry.com/p-articles/flower-of-life.php


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Slide 5-1: Pompeii pavementCalter 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, thepolygons,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

EquilateralTriangleTilings

Hexagon & Hexagram

Pentagon & Pentagram

Golden Triangle

Conclusion

Reading
https://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Polygonshttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Polygonshttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#EquilateralTrianglehttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#EquilateralTrianglehttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Tilingshttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Tilingshttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#HexagonHexagramhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#HexagonHexagramhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#PentagonPentagramhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#PentagonPentagramhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#GoldenTrianglehttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#GoldenTrianglehttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Conclusionhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Conclusionhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Reading5https://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Reading5https://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Reading5https://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Conclusionhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#GoldenTrianglehttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#PentagonPentagramhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#HexagonHexagramhttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Tilingshttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#EquilateralTrianglehttps://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Polygons


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Projects

Polygons

Slide 5-23: Design at Pompeii

Calter photo

In the last unit,Number Symbolismwe saw that in the ancient world certain numbers

had symbolic meaning, aside from their ordinary use for counting or calculating. Buteach number can be associated with a plane figure, orpolygon(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 assacred

geometry.

Apolygonis a plane figure bounded by straight lines, called the sides of the polygon.

From the Greekpoly= many andgon= angle

The sides intersect at points called the vertices.The angle between two sides is called

an interior angle orvertex angle.

Regular Polygons
https://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Projects5https://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Projects5https://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Projects5https://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html#Projects5


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A regular polygonis one in which all the sides and interior angles are equal.

Polygons vs. Polygrams

A polygramcan be drawn by connecting the vertices of a polgon.Pentagon & Pentagram,

hexagon & hexagram, octagon & octograms

Equilateral Triangle

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 withsides 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 whatthey 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


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of the Pythagorean brotherhood. We've seen it before in theSchool of Athens.

Trianglular Architectural Features

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.


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Slide 5-3: Greek Triskelion: Victory and ProgressLehner, 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.

Slide 5-5: Calter carving

Mandala II

Calter photo


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Slide 5-6: Closeup of wheel

Calter photo

Tilings

Slide 5-7: Pompeii Tiling with

equilateral triangles

Calter photo

Tilings or tesselationsrefers 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 tilingslike the Getty pavement shown here.


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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 DavidKeller, 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 regulartilings, because they use more than one shape.


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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 diamonddifferently.

Slide 5-10: Basket

Calter photo


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Slide 5-11: Pavement, DucalPalace, 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. SnowCrystals.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.


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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 asix

petalled rose.

Slide 5-13: Moses Cupola. S. Marco,

VeniceDemus, 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 calledthe 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


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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.

Slide 5-17: The genii emerging.Burton, Richard. The Arabian nights

entertainments.Ipswich : Limited Editions Club,1954.

The U.S. Great Seal

Slide 5-20: Seal on Dollar Bill

Calter photo


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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.

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.


25/30

Slide: 5-23. Design at PompeiiCalter photo

Slide 5-24: Design on Pisa Duomo

Calter photo

This hexagram is one of countless designs on the Duomo in Pisa.

Pentagon & Pentagram


26/30

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 Chiosis 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


27/30

The pentagon and pentagram are also interesting because they are loaded with Golden ratios, asshown 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 baseis the goldenratio.

It is also an isoceles triangle whose ratio of base to legis 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."


28/30

Penrose Tilings

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 TilingA Penrose tiling is made of two kinds of tiles, called kitesand darts.A kite is made

from two acute golden triangles and a dart from two obtuse golden triangles, as shown

above.


29/30

Slide 5-29: NCTM Cover

ConclusionSo 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!

Reading

Joseph Campbell, The Power of Myth,pp. 25-29

Carl Jung,Man andHis Symbols,pp. 266-285


30/30

Euclid,Elements,V2, pp. 97-104

Kappraff, Connections,pp. 85-87, 195-197

Fisher, p. 92-94

ProjectsCut 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

Fold an equilateral triangle using NCTM method

4

Construct a hexagon with compass 8

Construct a hexagon by paper folding, NCTM method 8Construct a hexagon by folding a circle 8

Make a pentagram by extending the sides of a pentagon, or make a

pentagram by connecting the vertices of a pentagon12

Construct a pentagon by either of Euclid's methods. Connect the vertices to

make a pentagram.13

Construct a pentagon by the Medieval method. Connect the vertices to make

a pentagram14

Construct a pentagon by Durer's method 14

Check for in the pentagon by using dividers 14

Solve the five-disk problem, Huntley p. 45 14

Put one type I and two type 11 golden triangles together to form a pentagon 15

Construct a triangle by Euclid's method 16

Construct a kite and a dart. Make xerox copies. Use them to make a Penrose

tiling.17

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