Through painstaking excavation archaeologists cast light on an ancient civilization from its material artifacts and their distribution. But ancient civilizations can teach us about themselves in other ways, too, including through studying the shapes and proportions of their sacred structures.
Because the Wright Place contains several unusual geometric mounds, much of the investigation of these mounds focuses on geometry. This page contains information about geometry in general, Native American geometry, and then explores some of the shapes present at the Wright Place.
Geometry
Geometry literally means “earth measure” and this term goes back to the ancient Egyptians who needed a way to reestablish property lines lost each year to the Nile’s annual flooding. But a tendency to think in terms of shapes and distances didn’t arise with the ancient Egyptians – some of the oldest cave art in the world incorporates abstract geometric forms.
Research with Amazonian tribes who have had very limited contact with modern society shows that they have an innate sense of connectiveness, understanding geometric shapes, angles and distances, suggesting geometry may be a result of how the human brain operates.
Many of the ancient Greek philosophers believed that geometry was so intrinsic to not just man but to the entire cosmos that rather than learning geometry, they maintained we “remember” it from a time before our birth. They believed numbers and shapes had dual natures, serving not only in the mundane sense as aids to counting and describing spaces but also as symbols of the ordered pattern of creation.
The Greeks believed that the external world of distances, shapes and proportions rested on an unseen geometric framework, a framework that unlike the ever-changing world it supports, is eternal and unchanging (our modern concept of unchanging physical laws underling the behavior of the Universe is an outgrowth of this view). Thus shapes and numbers came to be seen as the means by which God – also eternal and unchanging – created the phenomenal world, earning it the title of “Sacred” or “Creative” Geometry.
Finding geometry in the world around them led the Greeks and Romans to incorporate certain geometric ratios in their artwork and architecture, giving rise to some very beautiful creations and strengthening the belief that humans instinctively prefer “harmonious” geometric proportions. This ancient “canon” of shapes, numbers, harmonies and proportions was largely lost with the fall of Rome, but was reintroduced to Europe during the late Middle Ages and did much to inspire – and literally shape – the buildings, paintings and sculptures of the Renaissance.
Whether or not geometric shapes exist independent of the human mind, there’s no denying that the human mind finds patterns in the world around it, patterns that many cultures have recognized and used – probably often without thinking much about it. These cultures may not have devoted the intellectual energy to contemplating shapes that the Greeks did, but some form of geometry seems always to have been present. Even the simplest tribal dwelling tends to have a roughly symmetrical shape and one need only look at a triangular stone projectile point created anywhere in the world to realize that people interact with their environment through the use of shapes and sizes – the foundations of geometry.
Geometric Systems and Symbolism
It is no surprise then that cultures everywhere have depicted circles, squares, triangles and other shapes in their iconography. And like the Greeks, many other cultures attributed symbolic meaning to these shapes. And again like the Greeks, the symbolism given these shapes often had religious or spiritual meaning. As a result, certain geometric shapes were utilized for both symbolic and practical reasons when building religious structures. In India such a system is described in the ancient Hindu Manasara Shilpa Shastra while the Chinese system – Feng Shui – has gained favor in the West as a way to create harmonious spaces.
Long before the time of the Greeks geometry was being used by cultures like the Mesopotamians and Egyptians (what could be more geometric that a pyramid?). Geometry was also well understood by the ancient Europeans who built Stonehenge and a myriad of other stone circles and alignments. While we know little today about what led to the building of these enigmatic structures, their importance to their builders motivated them to move heavy stones sometimes great distances to create them. That geometry guided their arrangement shows its importance to these ancient people.
Native American Geometry
Though it’s clear from both their earthworks and their artwork that Native Americans understood geometry, little is known about Native geometry, let alone the spiritual significance it held. What is known indicates it was based on proportions and derived from simple circles.
The Mound Builders
Some of the oldest surviving mounds, those at Poverty Point in Louisiana, are impressively geometric. About 1,500 years old and monumental in size, these mounds extend 3/4 of a mile from end to end and cover 37 acres. Yet they were built by pre-agricultural hunter/gathers. No one knows why building these mounds was so important, but it is clear that these people were accomplished geometers.
Another impressive geometric earthwork is located near Newark, Ohio, and includes the largest earthen enclosures in the world. Though 90% of this site has been destroyed, it originally covered 3,000 acres and was constructed about 1,000 years after Poverty Point by people known as the Hopewell. A 20 acre Observatory Circle adjoins a 50 acre Octagon which is carefully aligned with the extreme rising and setting points of the moon – its alignment is twice as precise as the alignments at Stonehenge. The sizes of these two enclosure and two others – a square and another circle – are interrelated in respect to their area and perimeters. A standard unit of measure – 1,054 feet – has even been identified.
These are just two of many examples of geometry (and astronomy) having been incorporated in Native American earthworks. The vast majority of the mounds that once dotted the American landscape no longer exist, victims of farming and other forms of development – n0 doubt an enormous amount of knowledge has been lost with them.
The Pueblo
The use of geometry was not confined to the mound builders. Pueblo sites dating to about 900 years ago – the same general time the large village on the Wright Place is thought to have existed – incorporate the “Golden Ratio” or phi in their layout. Though not necessarily an indication of a connection between here and the Southwest (though it is believed there was contact between these two regions at that time), the Golden Ratio appears in several places in the mounds on the Wright Place.
Geometric Symbolism
In his book Indians of the Plains, anthropologist Robert H. Lowie attempts to discover any underlying symbolism in the geometric patterns of Native American art. In describing painted skins, Lowie explains that geometric designs were the province of women (men painted realistic scenes, such as battles). Among the Dakota Sioux women’s robes often featured a frame around an oblong field, while men’s robes often featured concentric circles with pairs of isosceles triangles radiating outward, the triangles symbolizing feathers. Among the Comanches and their neighbors, framed hourglass figures, frequently painted in red but also in blue or yellow were popular (colors also held symbolic meaning, though it varied from tribe to tribe).
Hide cases containing ritual objects often featured geometric designs, as did the more common parfleche, a rawhide bag used to carry dried meat mixed with berries. Lowie notes that Plains Indians achieved a “distinctive style” featuring various combinations of “straight lines, triangles, rectangles and diamonds” (though curves were sometimes present also). A quick search of Pinterest reveals just how important geometric shapes were – and still are – in Native American painting and quillwork.
Meanings
Lowie’s book includes illustrations of abstract geometric designs used by the Dakota and the meanings attributed to these shapes. Only the symbols for a tipi and an arrow resemble the actual objects. So Native Americans not only used abstract geometric images in their artwork, they attributed codified symbolism to them as well – certain shapes had specific meanings that weren’t realistically depicted. This symbolism varied among tribes, though certain correspondences between form and meaning were shared among different tribes. Among these was an association of the diamond with the navel (or the eye). Like the rectangle or a green square enclosing a red and white square, the diamond could also symbolize life. The turtle, which was generally associated with female physiological functions, was often symbolized as a “U” shape or by a triangle.
Lowie refers to these standardized geometric shapes and their symbolic meaning as the Native’s “technical nomenclature.” Presumably the mound builders likewise symbolized everything from life and reproductive power to feathers and tipis through abstract geometric shapes.
The Cheyenne
In his extensive catalog of Plains societies and cults, Dog Soldiers, Bear Men and Buffalo Women, Rev. Thomas E. Mails explains that the quill work decorating ceremonial objects – objects often created to fulfill and pledge or symbolize a vision – were created by women who belonged to a “craft guild.” Great ceremony accompanied the creation of these ritual objects, and Mails explains “The guild members had strict rules in their designs and they kept secret the meaning and arrangement of the colors, as well as the relation of the designs to each other. The designs were always symbolic and talismantic, representing concrete organic objects, whereas the colors were more emblematic of the abstract in creatures and creation.”
Because women worked with geometric shapes in their craft work, it is possible that the true meaning of these designs was, as in the case of the Cheyenne, was kept secret by the women who used them.
The Omaha
Like most tribes, the Omaha kept items used for performing rituals in sacred bundles. When it became undeniable that the Omaha would never return to their old ways of living, the hereditary keepers of some of these bundles gave them to anthropologists for safe keeping. Upon opening the sacred bundle kept in the tent of war, among other things, a geometric “flag” was found, bearing four rectangles with very precise geometric divisions. Though no one then alive knew anything about this flag, it was an important part of these people’s sacred artifacts.
Cultural Comparisons
The ancient Greeks, like every civilization, tried to explain the world around them and by eventually embracing a systematic approach based in part on mathematics and geometry, they laid the framework for modern science. Their science, however, was very much enmeshed with their religion, as it probably was with Native Americans also. Thus geometric forms and the mathematical ratios governing them may well have had sacred significance in the Native Americans, just as they’ve had in many other parts of the world, including ancient Greece.
In the section below I am using examples from Western culture to discuss certain universal geometric shapes that are found in all cultures. It is important to remember, however, that the examples given are not from Native American culture. It can be useful to compare different cultures to discover constants to human thought and action, but it can also be dangerous if one applies the literal meanings of one culture to another.
The Golden Mean or Phi Ratio
In our exploration of the geometry of the mounds William and I have encountered the “Golden” or Phi Ratio in several places. It was the ancient Greeks that named this ratio “golden” – presumably it had another name(s) among the Native Americans. But it appears to have been known to both the Greeks and the Native Americans – and many other cultures as well – and considered important in all of them.
The ancient Greeks loved to compare things through analogies – A is to B as B is to C, for example. Most ratios include four terms: A is to B as C is to D. So when things could be compared in only three terms, the Greeks took notice, feeling that this was a truer form of analogy (to the Greeks Jesus was the “logos”, the middle term in a three-term spiritual ratio with God as the first term and mankind as the third – that’s how important this form of thinking was to them – no relationship between extremes would be possible without the “mean” or middle term relating both extremes, whether the sides of a rectangle or God and man, to each other).
The Golden Ratio is a three term ratio, and is significant because of the unique way a line segment can be divided according to it.
A line segment can be divided in an infinite number of ways, and except for when it is divided exactly in half, one side of this division is always shorter than the other. If we call the shorter side of a divided line “A”, the longer side “B” and the full length of the line segment “C”, there is only one proportion that relates A to B to C in the same ratio or proportion. This is the Golden Ratio, in modern times denoted by the Greek letter phi. When a line segment is divided according to this ratio, A is to B as B is to C – in other words, if the length of the short segment (A) is divided by the length of the longer segment (B), the resulting ratio – which, like Pi never repeats and never ends – is the same ratio that results when the longer line segment(B) is divided by the length of the full line(C).
Mathematically this ratio is often expressed two ways – 0.618… to 1 or 1 to 1.618… Consequently phi can refer to either 0.618… or 1.618… Phi is intimately connected with the number 5, and is numerically equal to either (the square root of 5 +1)/2 = 1.618… or (the square root of 5 – 1)/2 = 0.618…
The phi ratio is found many places in nature, from the spiral shell of a chambered nautilus to the arrangement of seeds in a sunflower head (it even describes mathematically the reproductive rates of rabbits through something called the Fibonacci Sequence).
There are a lot of interesting – and unique – mathematical qualities of phi, but first and foremost it is a ratio between two different lengths. Thus it is prominent in geometry, determining the proportions of certain triangles, rectangles, rhombuses and pentagons.
Golden Rectangles
To construct a golden rectangle one begins with a square. The next step is to draw a diagonal from the bottom center point to the upper right (or left) corner. This diagonal is then swung down to where it aligns with the base of the square. A golden rectangle can then be created.
This ratio is created from a square:
Another unique feature of golden rectangles is that by adding a square (or subtracting one) from the long side creates a new golden rectangle –
Golden Rhombuses
A Golden Rhombus is a diamond shape created inside a golden rectangle by connecting the center points of each side.
A Golden Triangle is an isosceles triangle where the length of either side is in the phi ratio to the base. There are several ways to create a golden triangle, perhaps the simplest being to begin with a golden rectangle and swing the two long sides together to meet in the middle.
In the case of the golden rectangle the length of the shorter side to the longer side is 1 to 1.618. In the case of the golden triangle, the length of the base to either side (the longer sides being equal) is also 1 to 1.618.
We are all familiar with golden triangles – since the Golden Ratio is mathematically connected to the number 5, the five points of a star shape or pentagram are all golden triangles.
Golden Gnomon
Another type of Golden Triangle, the “Golden Gnomon,” can be created by placing two golden triangles point to point and rotating them so their top lines are straight. When their bottom points are joined a new triangle, a golden gnomon, is created –
In the golden gnomon triangle, the ratio of the side length to the base length is the Golden Ratio.
Pentagons
A pentagon begins life the same way a golden rectangle does, as a square divided vertically down the middle. Only now arcs are swung in both directions, creating two overlapping Golden Rectangles –
Next a new arc is swung from the lower right corner of the central square to the far left end of the arc (from points C to A) up to point E. This process is then repeated on the other side with points B and D –
Next, lines equal in length to the sides of the original square are drawn from B to F, and from C to G. Lines this same length are then drawn from F to E and from G to E, completing a pentagon.
Sacred Spaces
“Sacred” or “creative” geometry is a symbolic attempt to express the process by which the unseen becomes manifest, the process of creation. A belief that the manifest world is ordered by intangible geometric principles implies a belief in an unseen world, a world of spirit or unseen forces that precede and guide the development of physical forms. Often temples are seen as a microcosm of the universe so because geometry in this sense symbolizes the process of creation, it has been used all over the world in the design of sacred spaces.
Directions and Shapes
Humans have an innate sense of direction based on the fact that wherever you happen to stand there is something in front of you, on each side of you, and behind you. Thus we have the four directions of the world and because of this the Earth is often symbolized geometrically as a square. Juxtaposed with this is the unbounded circumference of a circle, a symbol associated with the heavenly realm, the realm without beginning or end. Thus the geometric squaring of the circle symbolizes the reconciliation of heaven and earth and has been a major pursuit of dedicated geometers throughout the ages.
Squaring the Circle
Squaring the circle is not as simple as it may sound. A circle and a square with the same perimeter won’t have the same area, and vice versa. Plus, there’s no simple way to created a square of either the same perimeter or area of a circle.
Sometimes the square and the circle are reconciled by simply drawing one within the other – either a circle with a square inside it that just touches the circle’s edge, or a square with a circle inside it that just touches the square’s edge.
The Octagon
Other times the heavenly circle and earthly square are reconciled through the 8-sided octagon, a sacred shape found in all baptismal fonts. There are several ways to create octagons, one of which is called the ‘Sacred Cut.’ This process begins with a large square and creates an octagon inside of it.
The first step is to find the center of the large square (the crossing point of the red lines in the diagrams). Then an arc is drawn from one corner whose length is equal to half the diagonal of the square. This process is then repeated from the remaining three corners –
The next step is to connect the points where the arcs touch the sides of the original square. The intersections of these lines create the inner, “sacred cut” square. Next the points where the arcs touch the sides of the original square are connected, forming four diagonals in the four corners. This creates an octagon inside the original square.
The image below shows on the left that if circles are drawn, one around the original square and one around the octagon, two of the major circles (red and green) that appear to help determine the dimensions of the Wright Mound are created –
The image on the right shows that the innermost blue circle is also related to the middle green circle through octagonal geometry. If a second octagon is created with the points touching the centers of the sides of the first octagon, the blue circle exactly fits inside.
9 to 10
In regards to squaring the circle, though none of these circles and squares match each others’ perimeter or area, there is a very interesting correlate to be found regarding the inner two circles (green and blue). If a square is drawn inside the green circle whose corners just touch the circle, the perimeter of this square (as measured in pixels with Photoshop) equals 9/10 the circumference of the blue circle –
The distance between the Yard Mound and the Rectangle Mound appears to be 9/10 the distance from the Rectangle Mound to the Wright Mound (all three mound are straight east/west of each other), and as explained on the Oval Mound page, this ratio of 9 to 10 is implied in the dimensions of the east oval. This 9 to 10 ratio appears to be the relationship the builders saw between 2 x the square root of 2 and pi (3.142 x 0.9 = 2.8278 ~ 2.828 — 2. 828 = 2 x the square root of 2).
Half the perimeter of the orange square (the measure of any two of its sides) would thus equal the square root of 2 in the builders’ system of symbolism. This is also the length of the diagonal of the larger, blue square that encloses the green circle, the width of which matches the width of the mound. It is also the diameter of the larger, red circle, the circle that touches the three widest points of the Wright Mound.
There appears to be much symbolism in the geometry determining various aspects of the Wright Mound’s peculiar shape, and its complexity reaffirms that the builders were very well versed in geometry.
Additional Geometry
The image below shows that these two octagons may determine some of the points on the waterline where the direction changes – notice how several points where the direction of the waterline changes align with lines drawn from the center through the corner of one of the octagons –
