# 5 compound tetrahedon - length of perpendicular lines between two edges and distance between these lines

The 5 compound tetrahedron:

is defined by 30 edges of equal length. Each of these edges pass by and do no intersect 4 other edges, giving us 4 pair of edges. For each pair we can construct a line segment that is perpendicular to both edges. These line segments will, if I'm not much mistaken, all be of equal length, so we need only know one, to know all 60. I need to know the length of this segment and I need to know where they connect on the edge.

I've been thinking about this problem for a long time, for a small project I have in mind, but I must say it's beyond me. At least I don't know where to begin. I had rather resigned myself to constructing the shape and taking measurements, which is in itself a rather involved and not at all precise process, but then I thought better of it and decided to ask here.

I do hope I can learn something, but if the solution goes beyond simple trigonometry and perhaps the odd quadratic, I don't think that'll happen, just so you know that this may well be a case of: "I need an answer, please do the hard work for me." that so many loathe.

In any case, thanks for your consideration.

The information found here may or may not be of use.

Edit: I have made a somewhat accurate representation of the problem and though it rather confusing, I hope it can explain my point.

This is of course only a part of the whole, and I have removed everything but the most essential, as it was just to confusing to look at otherwise.

The 4 long lines and the long red line all represents edges (though the may not look it, they are the same length) of the complete 5 compound tetrahedron, and the 4 short red lines and the vertices they include, represent the data I'm interested in.

In this case I've set the side length of the circumscribed dodecahedron to 10, the result of which is that the length of the 4 short lines are 1.69, well 3 of them are, the last one is 1.72, but that of course is a result of bad modelling or rounding errors, as the symmetry would otherwise be broken. edit: new model assures me that 1.69 was indeed correct.

As for the distance between vertices along the red edge, they are as follows, measured from left to right:

edit: New and more accurate numbers:

6.44

4.07

1.86

4.07

6.44

It doesn't get more accurate than this, without actually being able to calculate it. Still, I suppose I actually got the numbers I needed.

I hope this further clarifies matters, though I suspect not.

• I don't understand the question, what does "each of these edges pass by..." mean? – user202729 Aug 24 '18 at 15:45
• @user202729 I interpret the question as: For each edge, find the four shortest line segments to each of the four tetrahedra that the edge is not part of. It is far from literally in the question, so Zacariaz could you please clarify? – Inactive - avoiding CoC Aug 24 '18 at 15:51
• In the above, I mean the shortest segment from an edge to an edge of another tetrahedron. And for starters: By symmetry all these segments are indeed equally long. – Inactive - avoiding CoC Aug 24 '18 at 15:57
• I do apologize for not being clear enough, but I really don't know how else to explain it. I suppose my mathematical lingual skills leaves something to be desired. – Zacariaz Aug 24 '18 at 19:23