Pyramids, theirs Cross Sections Similar to Ellipses.

On this page we’ll consider only regular pyramids such that bases are regular polygons of many sides. A picture of such pyramid is shown above. The base of this pyramid is a regular polygon of 24 sides. Large number of sides causes that the base looks like a circle and that is why whole pyramid looks like a regular cone. We know that a cross section of a cone with a plane above the base is an ellipse.

This suggests that such cross sections of the described above pyramids remind ellipses. In fact it is so.

Let us we make the faces of the above pyramid transparent and let us cut the pyramid with a plane.

The above cross section looks like this (magnified):

May be it is not very well seen, but there is an ellipse drawn around cross section. This ellipse is very close to the perimeter.

This cross section is a polygon of 24 sides. We’ll try to describe it better showing its properties.

To each vertex there is corresponding “opposite” vertex (between them are 11 vertices either way). The next picture explains the definition. Pairs of “opposite” vertices are joined with a segment:

We see that these segments pass through the one common point. This is an obvious statement – this point is just a point where the height of pyramid crosses plane of cross section.

This point looks to be a focus of an ellipse. However, calculations show that it is not so. To check it you can use a criterion from a page of ‘Literka’ 'Ellipse as Planet's Trajectory' for a point to be a focus of an ellipse.

Let us proceed in a different way. Let us join with a segment all vertices except of “opposite” ones using different colors:

The obtained picture shows lot of regularity and many virtual ellipses. That is why people are using it for decoration purposes. They don’t just frame such pictures.

This is what they do:

Take a piece of a board. Cover it with a material. Hammer 24 nails at the points where vertices of the above polygon are (polygon must be magnified appropriately). Wind around these nails a thread. Use different colors of threads – color of segment of 2 vertices A and B depends only on the number of vertices between A and B. Thickness of a thread must be appropriate to the size of polygon – if a thread is too thick, virtual ellipses will not be seen.

It was noticed in the handbook of H. Steinhaus “Mathematical Kaleidoscope” that in the case of 23 vertices you can do it with only one piece of thread, without breaking it – but then of course everything will be in one color only. H. Steinhaus claims that winding of a thread takes about 45 minutes.

Return to the main geometrical page of 'Literka'.

Return to the main page of 'Literka'.