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This question has been edited.

The regular tetrahedron is a caltrop. When it lands on a face, one vertex points straight up, ready to jab the foot of anyone stepping on it.

Define a caltrop as a polyhedron with the same number of vertices and faces such that each vertex is at distance 1 from most of the corners of the opposing face. Are there any other caltrops besides the tetrahedron?

Use these 5 values in the list of vertices that follow. $\text{C0}=0.056285130364630088035020091792834$
$\text{C1}=0.180220007048851841582537343751297$
$\text{C2}=0.309443563867344767680227839435148$
$\text{C3}=0.348675924605445651138054435209609$
$\text{C4}=0.466391197450500551933366795454853$

verts =(
(C1,C1,C4),(C1,-C1,-C4),(-C1,-C1,C4),(-C1,C1,-C4),(C4,C1,C1),(C4,-C1,-C1),
(-C4,-C1,C1),(-C4,C1,-C1),(C1,C4,C1),(C1,-C4,-C1),(-C1,-C4,C1),(-C1,C4,-C1),

(C3,-C0,C3),(C3,C0,-C3),(-C3,C0,C3),(-C3,-C0,-C3),(C3,-C3,C0),(C3,C3,-C0),
(-C3,C3,C0),(-C3,-C3,-C0),(C0,-C3,C3),(C0,C3,-C3),(-C0,C3,C3),(-C0,-C3,-C3),

(C2,C2,C2),(C2,-C2,-C2),(-C2,-C2,C2),(-C2,C2,-C2));

The resulting polyhedron has the following appearance, arranged so that each of the three types of faces is on the bottom:

28-sided caltrop

Here's a transparent picture showing the 48 unit diagonals.

unit diagonals of 28-sided caltrop

Diagonals $(13-16, 14-15, 17-19, 18-20, 21-22, 23-24)$ have a length of about $~0.98620444$. I'm not sure of the maximum length, and don't have exact values for coordinates.

That's one more caltrop. Are there any others?

Rahul pointed out that some faces of my initial caltrop weren't exactly planar. This new version fixes that error, but I had to sacrifice 6 unit diagonals. A stronger caltrop would have each vertex at distance 1 from all corners of an opposing face, instead of most corners.

Ed Pegg
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  • Are you sure all the faces of your polyhedron are planar? I believe the bottom-left two and bottom-right two in the leftmost image (i.e. the ones a knight's move away from the triangle) are not. –  Jan 07 '16 at 02:09
  • I've fixed the error mentioned by Rahul. – Ed Pegg Jan 07 '16 at 19:30

1 Answers1

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There is a caltrop on 76 points.

76 point caltrop

Points 1, 13, 25, 29, 41, and 53 are as follows:

{0.0833`, 0.0833`, 0.4930122817942774`}  
{0.32530527130128584`, -0.20709494964790603`, 0.32530527130128584`}  
{0.28875291001058745`, 0.28875291001058745`, 0.28875291001058745`}   
{-0.2142`, 0.40369721678726284`, -0.2142`}  
{-0.07272969962634213`, 0.35355339059327373`, -0.35355339059327373`}  
{0.07587339432355446`, 0.44185`, -0.23402687345687453`}

This caltrop generates a solid of constant width. The polyhedron has 150 unit diagonals, is self-dual, and has tetrahedral symmetry.

Ed Pegg
  • 21,868