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ELLIPSOIDAL GYRATIONS OF PEACH


Introduction

This system was born out of my efforts to understand the properties of a small form which had been modelled by eye and needed enlarging to create a large stone carving.

The modelled form had a certain 'ring' to it, suggesting an underlying relationship of qualities intrinsic to its overall harmony.

The process of deciphering these relationships came after a long and frustrating period of largely fruitless efforts to re-model the form, which had originally been made over many winter evenings continually trying to perfect, in an intuitive sub-conscious way.

This is how I arrived at the beginning of a much longer logical process whereby I was to rigorously analyse the form and finally resolve its mysteries.

With very special thanks to Martin Tillett, with whom the latter part of the system was formed and without whose help it would not have been possible.

FEB'96


The System

This system can be summarised as a process starting in the third Dimension as a Sphere, being reduced to the second Dimension as a Circle, then morphed to become an Ellipse.  After various transformations are completed within this Elliptical format the cross-sectional information is brought back into the third Dimension to deliver the resulting form.

The purity of this process, with all elements so closely related to the Sphere, offers a final form clearly exhibiting the appropriate physical properties; within a harmony of proportion.

The elliptical cross-section chosen is the product of an exponential curve following properties of the 'Golden Section' or Φ proportion; which is 1.618 or 0.618.  This Ellipse describes the axis plane of its Ellipsoid to be discussed later.




This Ellipse sits within a square, axis running corner to corner.  The bisecting line "X" intersects the Ellipse to give start and finish of the 'valley line'.  This bisecting line was suggested by lines created in development of the 'Golden Section' of Φ Ellipse.  Its advantages are:-
  1. Any relationship between line "X" and Ellipse can be measured vertically in aspect to the sides of the square.
  2. The start and finish of bisecting line "X" are positioned within the Ellipse so both Tangents (where Ellipse brushes vertical sides of square) are correct for forming the 'valley line'.
  3. In this aspect the 'valley line' is created by bisecting the height between line "X" and the Ellipse above.  The line is plotted along points measured to be half way between the two.  The Character of this plotted line is that of the Ellipse above the bisecting line "X" being rotated towards the viewer.



With all information required ready plotted on the 'axis plane' we now have to conceptually transform the Golden Section (Φ) Ellipse into its Golden Section (Φ) Ellipsoid.  This is in order to establish a profile upon the plane perpendicular to the axis plane, at half the length of the Ellipsoid.  This plane gives the greatest cross-sectional circle within the Ellipsoid.




The profile placed upon 'Greatest circle plane' will be rotated about an axis within its plane, transforming as it turns; not only stretching elliptically to meet the Ellipsoidal space perimeter, but also under the influence of the "valley line" previously plotted upon the Ellipsoid's "axis plane".

The profile is derived from the characteristics of a double soap-bubble.  Two precisely equal sized spheres of film with a partition wall composed of the same substance all alike in contact with the air will meet at co-equal angles of 120.

This profile is drawn by dissecting the diameter line of a circle into three.  Using the two division points on the diameter line as foci an arc is established from each which intersect at line "A" (Ellipsoidal axis plane); below the diameter line remains as original circle.




Now we have our three main elements by which the eventual form is determined:-
  1. A defined Ellipsoidal space perameter.
  2. A 'valley line' plotted on the plane which bisects the Ellipsoid along its axis.
  3. A profile perpendicular to the Ellipsoid's axis plane, which, above its diameter line has the characteristics of a double soap bubble and below remains the original circle; and is located at half the length of the Ellipsoid.
So how do these three elements combine to deliver the final form?

When rotated the profile will transform to the appropriate Elliptical section of our Ellipsoidal space, stretching both the circle below its diameter line and shoulders of the soap bubbles above.

The shoulders above the diameter line of profile not only have to correspond to their appropriate Elliptical cross-sections but also to the side views "valley line", where the soap bubbles shoulders will eventually intersect.  The shoulders of the soap bubbles always remain above the side view bisection line "X".

To gain the correct Elliptical cross-sections of our Ellipsoid we refer to the side view diagram:




Here one can measure the distance from the centre of our original Golden Section (Φ) Ellipse to where a radial line intersects that Ellipse.  These cross-sections of the Ellipsoid are chosen for the practical reason of being evenly spaced between the start and finish of the "valley line".  This radial measurement gives us the half length of the transformed Elliptical section; the width remains the same.

Once the length of the new Elliptical section is known the whole Elliptical section can be gained by calculating the proportional difference between the length of the original Golden Section (Φ) Ellipse and the length of our new rotated cross-section.

Points around the original Ellipse can be measured vertically from its diameter line.  These measurements are then multiplied by the proportional difference in length of the two Ellipses.




We now have the possibility to find as many cross sections of the Ellipsoid as desired.

Of all possible Elliptical cross-sections there are two which transformation will not take place; the "Axis Ellipse" and the "Circle Ellipse" (they remain unchanged as they are the height and width of the Ellipsoid).

All Elliptical cross-sections can now be plotted, but the double soap bubble part of the profile (above its diameter line) as yet is not transformed.

To facilitate this next step let us take an example of one particular cross-section of the Ellipsoid.




Using the side view diagram we can measure the distance from the centre of our Elliptical cross-section to where the Valley is intersected by using our radial line as employed for finding the Elliptical cross-section.  This length is transformed to the profile diagram and measured vertically from the diameter line.  As expected this point differs from the original intersection point of our double soap bubble.

Transforming the original 'double soap bubble profile' (above its diameter line) to conform with the new positioning of intersection can be obtained as follows:-


(1)   Vertical measurement from chosen ellipse to its deepest point of valley 'X'   Which is the vertical distance between chosen ellipse and its shoulder point
=
(2)   Vertical measurement from original (3)   Vertical measurement from circle (at any point of arc above diameter line) to the shoulder point of its original soap bubble

So, to gain 'X': (1) is divided by (2) and multiplied by (3).




Here are some cross-sections of our resulting form.  Note that in this example the valley line has been plotted at one third of the distance between the Φ Ellipse and line X.
 
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