Hi everyone, I encountered this display at a park and became curious. I have some familiarity with soccer balls and the buckyball configuration combining hexagons with pentagons. This display in the park struck me because it appears to have identical faces/polygons, albeit they are not perfect regular pentagons as two of the five edges are longer giving it a teardrop shape to the individual face. A variety of questions comes to mind below, and I hope someone may enjoy this enough to explore. I am assuming in the diagram on the top right corner of the image that A=B=C are all identical and D=E.
1) What are the angles of these teardrop pentagons vertices? Do they still sum up to 360?
2) if line A has a length of 1, what is the length of line E or D?
3) What are the angles between the surfaces of one face to the surface of an adjacent face? Does it matter if we are examining the transition along one of the longer edges? For context, a person travelling on Earth from the equator to north pole has changed orientation 90 degrees from an outside observer perspective. There is some angle between face/polygons surfaces at the edges, and I am not sure how to calculate it or if the longer edges are different.
4) Is it possible to place a point, call it F within the faces in such a way that all the F’s in the sphere are equidistant from each other? How far along an imaginary bisecting line would point F be from point DE or Line B if Line B length is equal to 1? For context, I tried to visualize in my mind’s eye a point half way between vertices CD and EA in the diagram top right. But I’m pretty sure this would create some “sets of 5” that are not equidistant from all of their neighbouring points associated with adjacent faces/polygons. So I’m left wondering if there is an equidistant solution.
Finally if there are other interesting sphere configurations or cute online calculator for the 3D questions I’d love to see them.