Tag Archives: geometry

A Few Thoughts about Circles

Most nomadic tents are circular, the circle being the simplest, stable structure and also arguably the most beautiful, no corners to break up the continuity of curve, and for the energy to get stuck. The Native Americans have a saying that ‘the devil lives in corners’, probably originating when they were forcibly moved out of their tipis, into square stone houses on reservations.

Every time someone enters a yurt for the first time, and I still have this even after 14 years of making them, there is what I call the ‘cathedral moment’, when entering the circular space and the eyes rising to the central roof wheel with the sky opening above.

yurt wheel

And as the late Bill Coperthwaite, the first person to make a yurt in the West, said, “there is something indescribably beautiful about the view from a round window”.

yurt windows

Obviously in yurt-making we work with circle geometry a lot. At the simplest level, we calculate the circumference from the radius or diameter (2πR or πD), and floor surface area (πr2). We work out the spacing of gaps in the wheel based on the number of rafters, size of yurt based on the gaps between the trellis tops and the number of rafters, window placing with numbers on a clock etc (There is a lot of backwards and forwards between metric and imperial measurements as we talk about a 20 ft yurt but work in metric so to get from feet to metres divide by 3.28 and from metres to feet x3.28).

Why do we find circles so inuitively beautiful? In a 1921 study conducted by the Swedish psychologist Helge Lundholm, subjects were asked to draw lines representing a set of emotional adjectives. While angular lines were used to depict adjectives like hard, harsh and cruel, curved lines were the popular choice for adjectives like gentle, quiet and mild.

And what about the deeper beauty? Once upon a long ago there live Euclid, of Ancient Greece (c. 300BC), who was known as the ‘father of geometry’. His treatise ‘Elements’ is one of the greatest works in the history of mathematics, and is a collection of mathematical definitions, theorems, geometrical constructions, and mathematical proofs. In it he defines a circle as:

Definition 15 (abbreviated): ‘A circle is a figure contained by one line (the circumference) such that the length of all the straight lines (the ‘radii’) falling on it from one point (the ‘centre’) equal one another.

This is a definition as beautifully smile (that should have been ‘simple’ but spellcheck gave me ‘smile’ !) as could be, but mathematicians are extremely subtle and rigorous creatures so it should be noted that a definition such that as this describes what circles ARE, but definitions do not guarantee the existence of the things they define. We won’t however be wandering down that pre-existential corridor here.

From this basic definition, using just Euclid’s ruler and compass one can create endless geometrical beauty. The most basic is the equilateral triangle:

Construction of Equilateral Triangle using intersecting arcs

And (once you have constructed a square, also using only ruler and compass – not a trivial thing), we have Square in a Circle. (The lighter lines are construction lines).

Circle Geometry
Square in a Circle

And then Square around a Circle

Circle Geometry
Square around a Circle

And ….

A familiar construction, and ancient symbol, associated with sacred geometry is the seed of life, constructed from overlapping circles, which can be extended infinitely to the flower of life. This symbol has been of particular interest of us as the central structure of our Zodiac Tent, the helix zome, as when looked at from above, you will see the intersecting struts form the flower of life. Here is a blog post from the archives with some references to this.

Seed of Life

Flower of Life

Then, of course we come to the magical number pi, π, originally defined as the ratio of a circle’s circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics.

Most people know:

C=πD or C=2πR

Less well known is the formula for surface area of a sphere:

And Volume of a sphere is given by

The digits of pi are like gazing into the infinite as it is an infinite series, with no apparent pattern. Infinite series have fascinated mathematicians for centuries.

In 2004 an autistic savant by the name of Daniel Tammet set the record for pi memorization, memorizing 22,514 digits in just over 5 hours. A savant is someone with significant mental abilities, far in excess of average. These people are often defined as having a mental ‘disorder’ such as autism, but I think this definition lacks a magical element, that these people are tapping into a much greater psychic field than is usually accessible. He says he sees the numbers as complex, 3-dimensional landscapes, complete with color, texture, emotion and sound and journeying through this inner landscape, unfolding like a beautiful poem is how he memorises the numbers. I am fascinated by these incredible powers but Tammet explains that the differences between savant and non-savant minds have been exaggerated; autistic thought, he argues, is an extreme variation of a kind of highly rich and complex associative form of thinking and imagination, a kind that we we all use, from daydreaming to the use of puns and metaphors …… maybe if the consensual reality wasn’t as it is, and the education system wasn’t such a sausage factory, and the world was oh so different, more of these magical abilities would emerge, and would become a normal occurence. ‘(If anyone is interested in reading books by Daniel Tammet, who is quite a rare savant with extremely advanced abilities, who is able to describe his inner processes, I can recommend ‘Born on a Blue Day’, ‘Thinking in Numbers: On Life, Love, Meaning and Math’, and ‘Embracing the Wide Sky: A Tour Across the Horizons of the Mind’).

The circle represents the infinite.