Minimizing the Ratio of
Maximum to Minimum
Distance in 3 Dimensions

The following pictures show n points in space so that the ratio r of maximum distance (shown in red) to minimum distance (shown in blue) is the smallest known.
4.
r2 = 1
Trivial.
(regular tetrahedron)
5.
r2 = 12/7 = 1.714+
Found by David Cantrell
in March 2009.
(3-fold rotational symmetry)
6.
r2 = 2
Found by David Cantrell
in March 2009.
(regular octahedron)

7.
r2 = 2.293+
Found by David Cantrell
in March 2009.
(3-fold rotational symmetry)
8.
r2 = 1 + √2 = 2.414+
Found by David Cantrell
in March 2009.
(square antiprism)
9.
r2 = (9 + √129) / 8 = 2.544+
Found by David Cantrell
in March 2009.
(3-fold rotational symmetry)

10.
r2 = 3.101+
Found by David Cantrell
in March 2009.
(4-fold rotational symmetry)
11.
r2 = 3.384+
Found by David Cantrell
in March 2009.
(3-fold rotational symmetry)
12.
r2 = (5 + √5)/2 = 3.618+
Found by David Cantrell
in March 2009.
(regular icosahedron)

13.
r2 = 3.947+
Found by David Cantrell
in March 2009.
(asymmetric)
14.
r2 = 4.168+
Found by David Cantrell
in March 2009.
(asymmetric)
15.
r2 = 4.398+
Found by David Cantrell
in March 2009.

16.
r2 = 4.553+
Found by David Cantrell
in March 2009.
(2-fold rotational symmetry)
17.
r2 = 4.771+
Found by David Cantrell
in March 2009.
(5-fold rotational symmetry)
18.
r2 = 5.047+
Found by David Cantrell
in March 2009.
(5-fold rotational symmetry)

19.
r2 = 5.409+
Found by David Cantrell
in March 2009.
(4-fold rotational symmetry)
20.
r2 = 5.799+
Found by David Cantrell
in March 2009.
(asymmetric)
21.
r2 = 5.985+
Found by David Cantrell
in March 2009.
(2-fold rotational symmetry)

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