Geometry, Art and Nature

I have been playing with a compass and straight edge lately. It is relaxing and at the same time, thought provoking.  I was inspired by Paul Klee‘s work, even though it seems that his credo of ‘taking a line for a walk’ doesn’t seem to mesh with the strictures of the compass and straight edge.

Constructing circles, bisecting lines, connecting points on the circumference, is relaxing to the extent that one can see patterns emerging.

However, I found that attempting to create a specific figure can be maddening. The other night, when I had trouble sleeping, instead of counting sheep, I would try to figure out how to construct an equilateral pentagon inscribed in a circle. That was a mistake. First I had to figure out the angle at each vertex. That wasn’t too bad. I came up with 108 degrees. The difficult part was trying to figure out how to construct lines that intersect at that angle.  I couldn’t do it.

I thought of another approach: Instead of trying to construct the angle, figure out how to divide the circumference of the circle into five equal arcs. I couldn’t do that either.

I was up for hours mulling over these thoughts.

A hint from The Healing Garden gardener (THGg)

THGg provided me with a  reference to an animated construction of a pentagon inscribed in a circle (http://www.mathopenref.com/constinpentagon.html). I have to look at it in a little more detail, but I am not certain that this figure is a pentagon with equal sides.

Question

The thought occurred to me that perhaps it is not possible to inscribe an equilateral pentagon in a circle. So I worked backwards. Using a protractor I constructed an equilateral pentagon, with each of the five internal angles equal to 108 degrees. After finding the center of the figure by drawing lines between vertices, I was indeed able to draw a circle that touched each point on the pentagon.

I need to revisit this.

Thanks to THGg for his interest and follow up on his blog.

Today’s experiment

To a person with a hammer, the whole world looks like a nail.  This seems to be the case with today’s watercolor sketch.

I was in the kitchen and happened to look at a pineapple that was on the counter. The circular opening on its bottom was surrounded by triangular shapes. I thought, “Could nature have solved the inscribed polygon problem, proved by Gauss?”

Watercolor Sketch - Bottom of a Pineapple

Pineapple Bottom
12″x9″ 140# Cold Pressed Watercolor Block

This watercolor is a visual edit of the pineapple bottom. I was interested in the triangular shapes and how they were disposed around the circle that I assume was the attachment point of the pineapple fruit to the rest of the plant.

As in yesterday’s sketch, the shapes in this sketch are accurate, but the coloration and shading is a bit off. The triangular shapes at the top are translucent and a bit fluted, as if each triangle was a leaf with parallel veins. The pinkish color fades to yellow and brown at each vertex. The brownish color at the top of the inside circle could be more defined, as it represents the shadow of the recessed center. In addition, the triangular shapes at the bottom should be much brighter, as the light fell on them directly.

Future work

I would like to develop a method of working that combines aspects of compass and straight-edge construction with the freedom of Klee’s idea of ‘taking a line for a walk’.

 

2 thoughts on “Geometry, Art and Nature

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