Nested Parabolas

 
I asked myself what it would look like if parabolas were drawn on the backs of one another. I imagined something like the following:

Clearly, the arms of a second parabola would need to be longer than those of the first. How short could these arms be if the two parabolas intersect only at their endpoints? With some simplifying assumptions, we can answer this question.

Assume the center point of a parabola is at the origin, and the endpoints are at A = (-w, d) and B = (w, d). Thus, we can draw a parabola with arms of equal length. (Think of d as the depth of the parabola and 2w as its width.) We want to draw a second parabola with the same endpoints and a center point at B = (0, x), where x is negative. The construction needs to be such that none of the lines of the parabola cross the y-axis above the origin, where such a line would intersect the center point of the first parabola.

It is easy to see that, if a parabola is drawn using the arms AB and BC, the line of the parabola that intersects the y-axis as high as possible will be the line connecting the midpoints of AB and BC. We can minimize x by having the line intersect the origin, that is, the center point of the original parabola. To connect these points |x| must be equal to d. (Whether the lower parabola contains a horizontal line connecting the midpoints of its arms depends on whether the number of divisions on the arms is odd or even.)

Generalizing the size of the n^th parabola is less straightforward. The logic is basically the same, the depth of the second parabola is twice that of the previous parabola. Therefore, the center point of the next parabola is at (0, 4d). The center point of the n^th parabola is at (0, ‑(dⁿ)).

With that out of the way, we can begin to create some figures. Here is a sample combining four parabolas:

Some observations:

• The size of the figure grows quickly as parabolas are added.
• Increasing d increases the size of the figure.
• If the arms of each parabola are divided into the same number of segments, the length of the segments increases with each parabola added. (We will return to this phenomena below.)

The first “interesting” construction I developed put two parabola nests back-to-back:

My design became more interesting when I combined them with (invisible) polygons. Here is an example:

We can manipulate the number of nested parabolas, the number of segments into which parabola arms are divided, the number of sides of the polygon, and the width and depth of the parabolas. Here is an example in which all of those parameters are modified:

Here is another design:

Notice that the lines of the innermost parabola are denser than those of the outermost parabola. We can, if we like, maintain the same or nearly the same spacing of arm segments. This works out well in some cases, as in this design:

Adjusting the segments to be of equal length everywhere may or may not seem a good choice because later parabolas seem much denser than earlier ones. Here is another figure in which the segment lengths have been homologized. Decide for yourself how well this works in the figure:

Rather than keeping all segments the same length, one could make adjustments that made spacing seem uniform without actually being uniform. I have not experimented with this idea.

Perhaps other sorts of figures could be made with nests of parabolas as shown above. I leave it to others to discover such designs.

 
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