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 = 4.165+
Found by M. Sun and E. Samanta
in June 2026.
(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)

22.

r² = 6.345524+
Found by M. Sun and E. Samanta
in June 2026.
(asymmetric)

23.

r² = 6.613191+
Found by M. Sun and E. Samanta
in June 2026.
(asymmetric)

24.

r² = 6.901318+
Found by M. Sun and E. Samanta
in June 2026.
(asymmetric)

25.

r² = 7.045279+
Found by M. Sun and E. Samanta
in June 2026.
(asymmetric)

26.

r² = 7.305305+
Found by M. Sun and E. Samanta
in June 2026.
(2-fold rotational symmetry)

27.

r² = 7.526055+
Found by M. Sun and S. Coutinho
in June 2026.
(asymmetric)

28.

r² = 7.689589+
Found by M. Sun and P. Chopra
in June 2026.
(asymmetric)

29.

r² = 7.849436+
Found by M. Sun and R. Mattankot
in June 2026.
(asymmetric)

30.

r² = 8.031711+
Found by M. Sun and A. Shah
in June 2026.
(3-fold rotational symmetry)