Pentagonal Pyramids

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Here is a set of Pentagonal Pyramids, built so that all pyramids are topped with 5 struts converging onto the apex, and the struts at the top, if extended, meet the corners of the base pentagon. So their five sides are "flat". The length of most struts is 1 unit. Gold means the golden ratio ( the ratio of pentagons' side to its diagonal ).

 

Simple Pyramids....

In the one on the right, all struts are the same length. The side of the base is 1, and the distance from the corner of the base to the top is Gold.
This pyramid is 5 squares rising up from the pentagonal base to form a pentagram, which forms the base of 5 equilateral triangles meeting at the apex.
Alternatively you can see 5 equilateral triangles rising from central pentagram, and also 5 equilateral triangles below! It is kind of stiff but I am not sure how rigid it is.

In the one on the left, all struts are the same length. As in the tall pyramid before, the distance from a corner of the base to the top is Gold. Also like the previous one, the uppermost struts meet the lower ones at points forming a pentagon, and the ratio of the sides of this pentagon to that of the base is Golden.

 

A Taller Pyramid..
Here, the struts on the "ground floor" are of length 1, and it turns out the length of the upper floor struts has to be Root(2). The ratio of the length of a lower strut to the side of the base is Golden, which explains the pentagram being on the base. The uppermost struts meet the lower ones at points forming a small pentagon I marked out in string. The ratio of the sides the base pentagon to this one is Golden.

To give this structure rigidity, 5 inner struts have been added, starting at the corners of the base, passing eachother in the middle, and ending up at the intersection of the main struts leading up from the base. These struts are length 1.

You can perhaps see this pyramid is a base pentagram forming the base for 5 equilateral triangles, which meet in such a way as to intersect eachother in Golden sections. The reinforcement struts which support these intersections also meet each other at a Golden cut of themselves.
If you look very hard perhaps you can see 5 Golden rectangles rising from the base pentagram.

On the right is the same pyramid with added strings to show an interior icosahedron, and also an extension of the lower parts to meet the top of this interior icosahedron. 

 

A Larger Pyramid..
The sides of this pentagonal pyramid are part of a very interesting pattern. More on that later.

To add rigidity, I supplemented the sides by adding horizontal "floors", and at the lower levels some extra sloping struts, whilst maintaining the symmetry.

All the struts in this framework are the same length.

One aspect of this pyramid I particularly like is they way the that it consists almost entirely of repeated squares, triangles, and "thin" and "fat" diamond shapes.

Unfortunately the best way to understand this pyramid is in 3d, so these 2d pictures dont really do justice.
Here you can see one side colored in pink and the floors each colored in their own color.

The ground floor is a blue pentagram inside a blue pentagon, with a star right in the middle.

The first floor is a yellow pentagram containing a yellow star.

The third floor is a blue pentagon.

The fourth floor is a yellow star, and the top is one point.

From this viewpoint you can clearly see some of the inner squares.
Here you can clearly see the layers between the floors.

I could have added extra floors, but I didn't, not for any particular reason. Perhaps I will at a later date.

 
Here it is lying on one of its sides!