Place n points in the plane or in space to minimize the squared ratio of the largest to the smallest pairwise distance. Values are exact rationals of the exported coordinates.
| Problem | n | Listed on the page | Our certified value | Margin | Margin % | Status | Listed by |
|---|---|---|---|---|---|---|---|
| Max/min distance ratio, plane | 9 | r^(2) = 6.600+ | 6.600978659 | −9.79e-04 | -0.015 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 10 | r^(2) = 7.713+ | 7.713455164 | −4.55e-04 | -0.006 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 11 | r^(2) = 8.222+ | 8.222295312 | −2.95e-04 | -0.004 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 12 | r^(2) = 5 + 2√3 = 8.464+ | 8.464101615 | −8.88e-15 | -0.000 | reproduced | Found by Ingo Rechenberg in 2006. |
| Max/min distance ratio, plane | 13 | r^(2) = 9.930+ | 9.930035789 | −3.58e-05 | -0.000 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 14 | r^(2) = 10.994+ | 10.994717906 | −7.18e-04 | -0.007 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 15 | r^(2) = 12.038+ | 12.038438278 | −4.38e-04 | -0.004 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 16 | r^(2) = 12.890+ | 12.889229908 | +7.70e-04 | 0.006 | reproduced | Found by Timo Berthold et al in January 2026. |
| Max/min distance ratio, plane | 17 | r^(2) = 14.090+ | 14.090518705 | −5.19e-04 | -0.004 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 18 | r^(2) = 14.725+ | 14.725276092 | −2.76e-04 | -0.002 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 19 | r^(2) = 8 + 4√3 = 14.928+ | 14.928203230 | −1.07e-14 | -0.000 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 20 | r^(2) = 16.734+ | 16.734525501 | −5.26e-04 | -0.003 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 21 | r^(2) = 17.774+ | 17.774980554 | −9.81e-04 | -0.006 | reproduced | Found by Timo Berthold et al in January 2026. |
| Max/min distance ratio, plane | 22 | r^(2) = 19.053+ | 19.053984510 | −9.85e-04 | -0.005 | reproduced | Found by Timo Berthold et al in January 2026. |
| Max/min distance ratio, plane | 23 | r^(2) = 20.039+ | 20.039612312 | −6.12e-04 | -0.003 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 24 | r^(2) = 14 + 4√3 | 20.928203230 | −4.26e-14 | -0.000 | reproduced | Found by Charles Audet et al in January 2010. |
| Max/min distance ratio, plane | 25 | r^(2) = 22.121+ | 22.121568203 | −5.68e-04 | -0.003 | reproduced | Found by David Cantrell in February 2009. |
| Max/min distance ratio, plane | 26 | r^(2) = 22.810+ | 22.810592955 | −5.93e-04 | -0.003 | reproduced | Found by Charles Audet et al in January 2010. |
| Max/min distance ratio, plane | 27 | r^(2) = 23.083+ | 23.083331496 | −3.31e-04 | -0.001 | reproduced | Found by Ingo Rechenberg in 2006. |
| Max/min distance ratio, plane | 28 | r^(2) = 24.981+ | 24.981438861 | −4.39e-04 | -0.002 | reproduced | Found by Charles Audet et al in January 2010. |
| Max/min distance ratio, plane | 29 | r^(2) = 25.924+ | 25.924603994 | −6.04e-04 | -0.002 | reproduced | Found by Timo Berthold et al in January 2026. |
| Max/min distance ratio, plane | 30 | r^(2) = 26.879+ | 26.879264785 | −2.65e-04 | -0.001 | reproduced | Found by Charles Audet et al in January 2010. |
| Max/min distance ratio, plane | 31 | r^(2) = 28 | 28.000000000 | −3.55e-14 | -0.000 | reproduced | Found by Gianluca Costantino in August 2026. |
| Max/min distance ratio, space | 5 | r^(2) = 12/7 = 1.714+ | 1.714285714 | −4.44e-16 | -0.000 | reproduced | Found by David Cantrell in March 2009. (3-fold rotational symmetry) |
| Max/min distance ratio, space | 6 | r^(2) = 2 | 2.000000000 | −8.88e-16 | -0.000 | reproduced | Found by David Cantrell in March 2009. (regular octahedron) |
| Max/min distance ratio, space | 7 | r^(2) = 2.293+ | 2.293437256 | −4.37e-04 | -0.019 | reproduced | Found by David Cantrell in March 2009. (3-fold rotational symmetry) |
| Max/min distance ratio, space | 8 | r^(2) = 1 + √2 = 2.414+ | 2.414213562 | −2.22e-15 | -0.000 | reproduced | Found by David Cantrell in March 2009. (square antiprism) |
| Max/min distance ratio, space | 9 | r^(2) = (9 + √129) / 8 = 2.544+ | 2.544727086 | −2.22e-15 | -0.000 | reproduced | Found by David Cantrell in March 2009. (3-fold rotational symmetry) |
| Max/min distance ratio, space | 10 | r^(2) = 3.101+ | 3.101626005 | −6.26e-04 | -0.020 | reproduced | Found by David Cantrell in March 2009. (4-fold rotational symmetry) |
| Max/min distance ratio, space | 11 | r^(2) = 3.384+ | 3.384171992 | −1.72e-04 | -0.005 | reproduced | Found by David Cantrell in March 2009. (3-fold rotational symmetry) |
| Max/min distance ratio, space | 12 | r^(2) = (5 + √5)/2 = 3.618+ | 3.618033989 | −3.11e-15 | -0.000 | reproduced | Found by David Cantrell in March 2009. (regular icosahedron) |
| Max/min distance ratio, space | 13 | r^(2) = 3.947+ | 3.947141175 | −1.41e-04 | -0.004 | reproduced | Found by David Cantrell in March 2009. (asymmetric) |
| Max/min distance ratio, space | 14 | r^(2) = 4.165+ | 4.165783475 | −7.83e-04 | -0.019 | reproduced | Found by M. Sun and E. Samanta in June 2026. (asymmetric) |
| Max/min distance ratio, space | 15 | r^(2) = 4.398+ | 4.398108949 | −1.09e-04 | -0.002 | reproduced | Found by David Cantrell in March 2009. |
| Max/min distance ratio, space | 16 | r^(2) = 4.553+ | 4.553202961 | −2.03e-04 | -0.004 | reproduced | Found by David Cantrell in March 2009. (2-fold rotational symmetry) |
| Max/min distance ratio, space | 17 | r^(2) = 4.771+ | 4.771073409 | −7.34e-05 | -0.002 | reproduced | Found by David Cantrell in March 2009. (5-fold rotational symmetry) |
| Max/min distance ratio, space | 18 | r^(2) = 5.047+ | 5.047608573 | −6.09e-04 | -0.012 | reproduced | Found by David Cantrell in March 2009. (5-fold rotational symmetry) |
| Max/min distance ratio, space | 19 | r^(2) = 5.409+ | 5.409960182 | −9.60e-04 | -0.018 | reproduced | Found by David Cantrell in March 2009. (4-fold rotational symmetry) |
| Max/min distance ratio, space | 20 | r^(2) = 5.799+ | 5.799520109 | −5.20e-04 | -0.009 | reproduced | Found by David Cantrell in March 2009. (asymmetric) |
| Max/min distance ratio, space | 21 | r^(2) = 5.985+ | 5.985519297 | −5.19e-04 | -0.009 | reproduced | Found by David Cantrell in March 2009. (2-fold rotational symmetry) |
| Max/min distance ratio, space | 22 | r^(2) = 6.34552+ | 6.345524236 | −4.24e-06 | -0.000 | reproduced | Found by M. Sun and E. Samanta in June 2026. (asymmetric) |
| Max/min distance ratio, space | 23 | r^(2) = 6.61319+ | 6.613190831 | −8.31e-07 | -0.000 | reproduced | Found by M. Sun and E. Samanta in June 2026. (asymmetric) |
| Max/min distance ratio, space | 24 | r^(2) = 6.90131+ | 6.901317674 | −7.67e-06 | -0.000 | reproduced | Found by M. Sun and E. Samanta in June 2026. (asymmetric) |
| Max/min distance ratio, space | 25 | r^(2) = 7.04527+ | 7.045279119 | −9.12e-06 | -0.000 | reproduced | Found by M. Sun and E. Samanta in June 2026. (asymmetric) |
| Max/min distance ratio, space | 26 | r^(2) = 7.30530+ | 7.305305322 | −5.32e-06 | -0.000 | reproduced | Found by M. Sun and E. Samanta in June 2026. (2-fold rotational symmetry) |
| Max/min distance ratio, space | 27 | r^(2) = 7.52605+ | 7.536738811 | −1.07e-02 | -0.142 | behind | Found by M. Sun and S. Coutinho in June 2026. (asymmetric) |
| Max/min distance ratio, space | 28 | r^(2) = 7.68958+ | 7.689588973 | −8.97e-06 | -0.000 | reproduced | Found by M. Sun and P. Chopra in June 2026. (asymmetric) |
| Max/min distance ratio, space | 29 | r^(2) = 7.84943+ | 7.849436186 | −6.19e-06 | -0.000 | reproduced | Found by M. Sun and R. Mattankot in June 2026. (asymmetric) |
| Max/min distance ratio, space | 30 | r^(2) = 8.03171+ | 8.034732554 | −3.02e-03 | -0.038 | behind | Found by M. Sun and A. Shah in June 2026. (3-fold rotational symmetry) |
| No cases match this filter. | |||||||
Data as of 2026-09-12 05:21 UTC. Every value is certified by an independent validator; none is a claim of optimality. Nothing has been submitted to the page.