Euler’s Cube – Fosbury 2010

On July 17^th 2010 near Vernham Dean, Wiltshire, UK the 250ft Fosbury crop circle was sighted. While many have looked at this crop through a 3 dimensional perspective linking its geometry to the Tesseract (or Hypercube), an even more intriguing perspective is hidden within a 2 dimensional analysis of the crop.

You can look at the crop as if it were a smaller cube nested inside a larger cube with some interesting nested divisions of the faces of the external cube’s faces. This is where some of the hypercube interpretations begin and end, really. However, if you take to analyzing the projections of the cubes onto the 2D plane, you begin to see nested hexagons.

After Adam Apollo performed his analysis on the hexagon (or cube) edge lengths, an interesting connection between the inner hexagon and outer hexagon edge lengths emerged. While the outer hexagon edge length to the edge length of the flattened crop gave Pi, the same outer hexagon edge length to the full “hidden” inner cube (including its border) gives Euler’s number!

EdgeLength_OUTER / EdgeLength_INNER ≈  e = Euler’s Constant
2.71318 ≈ 2.71828 

Now, if you were to nest a circle inside the perimeter of each of the INNER and OUTER 2D hexagons, this would be the same as nesting two spheres inside each cube. Where,

EdgeLength_OUTER = SphereDiameter_OUTER, and

EdgeLength_INNER = SphereDiameter_INNER

Bringing this back into the 2D space, we know the diameter of a sphere is equal to the diameter of its 2D projection of a circle so we can also say,

D_OUTER / D_INNER = e

Now that we can see the Euler relationship down in the 2D plane with circles representing spheres, we continue to nest circles inside the perimeter of the each of the remaining 5 hexagons present between the inner and outer hexagons. We find that each of the 5 circles have diameters that divide the space between the inner and outer circles by equal 12ths. If you were to fully divide the space by circles, their diameters would be

D_N = D_INNER + N*(D_OUTER – D_INNER) / 12
Where N = 1,..,12

From here we can begin to see a relationship emerge: a division of space in 12ths with a growth (or decay) rate of e.

Music theory and the Circle of 5ths is a great analogous tool to describe this infinite division of space. If we map the 12 axial divisions of the Circle of 5ths onto our circular map, we can more easily imagine the traversal of the 12 Major notes in the Circle of 5ths  (C, G, D, A , E, B, F#, Db, Ab, Eb, Bb, F) as if each note were mapped to each circle (or sphere). Now, we know that we can continuously traverse the Circle of 5ths into ever growing, or decaying  harmonic frequencies within audible range. We can visualize this traversal through the use of an Archimedes spiral connecting the edge of each successive circle to the 12 major axial divisions of the space. (See image below.)

We can further the visual exercise by repeating the pattern in either a growth or decay representation by repeating the pattern with another set of circles with each of their diameters scaled by a factor e. Outside the bounds of the analogy of audible frequencies, we can extrapolate a relationship between spheres, their diameters, and potentially their resonant frequencies.

We know that where Euler’s constant shows up, there is usually a natural growth or decay function being described that can often be measured in nature. Can this relationship credit Pythagoras for the discovery of music theory’s harmonious Circle of 5ths being yet another description of a natural growth and decay phenomena? Or perhaps this interesting relationship is one we can use to describe other physical phenomena in our 3D reality. It’s open to further investigation.

– Analysis by Carissa Ouellette