Heptagonal Domes

index

This is a heptagonal dome made out of seven leaning heptagons, whose bottom segments form another. In the picture, I have coloured two of the leaning heptagons as well as the base. The red triangle is equilateral and is the triangle formed by joining the corners of the heptagons as shown. There is more about this triangle in my regular polygons page.

The dome is given rigidity by a lemon hanging from the apex that pushes the sides out, and by nylon threads stretched horizontally at the nodes on the first and second floors to push the sides in.

• The base is a heptagon. Suppose each side is length 1.
• The "first floor" is an irregular polygon with 14 sides, seven of them are of the longer length d ( the length of the side of the equilateral triangle ), and seven of them are of the shorter length g.
• The "second floor" is a regular heptagon. Since the angles at the apex are the same as the angles rising from the base, it follows the horizontal lengths of the sides of the second floor are the same as d on the first floor.
• The apex is the meeting point of 7 of the leaning heptagons' corners. 

First, here is a picture of one of the leaning heptagons.

Note:

• The short diagonal b is the long diagonal of the top floor of the dome.
• The long diagonal c is one of the diagonals of the first floor of the dome.
• d is the side of the heptagon whose long diagonal would be b.

For the moment just notice the labelling of length g. We'll prove it is the short segment lenth of the dome shortly.

The Top Floor

Here is a diagram of the top floor, with d and b labelled.

The First Floor.

The segments of the first floor are labelled in pink and turquoise. The diagonals c and b as described above are shown in green and grey. Using similar triangles, you can see that the segments labelled d are indeed the same length. Also by similar triangles, we can see that the segments marked g are also the same length, and the same length as originally defined above.

So, we have defined the lengths d and g completely in terms of the original dimenstions of the leaning heptagons. The pictures show this geometrically but you could express it as ratios if you wanted. The only difficulty you would have is expressing their length with numbers. In fact, you would have a great deal of difficulty because the diagonals of a unit regular heptagon are not expressible as a normal formula with square roots and so on - but that is not what I am interested in so you should look elsewhere if you are into that kind of thing.

There is a lot more to heptagonal geometry than you would expect.