The Rupert property of a polyhedron derives from a mathematical question formulated by Prince Rupert of the Rhine in the 17th century. A polyhedron P is said to have the Rupert property if a polyhedron of the same or larger size and the same shape as P can pass through a hole in P.
Steininger and Yurkevich, in their paper An algorithmic approach to Rupert’s problem, found that 82 of the Johnson solids had the Rupert property, but didn't publish their parameters. Parameters for these solids were produced by running Johnson Solids Solver.R from the git repository accompanying their paper. Fredriksson reported in his paper The triakis tetrahedron and the pentagonal icositetrahedron are Rupert that five additional Johnson solids had the Rupert property. Parameters for these (J25, J45, J47, J71, J76) are included below, although the translations x and y have been changed slightly.
| name | x | y | α | θ1 | ϕ1 | θ2 | ϕ2 |
|---|---|---|---|---|---|---|---|
| J1 | 0.0447957 | -0.0168180 | 1.6692911 | 0.5948349 | 1.6223952 | 4.8778858 | 2.8291271 |
| J2 | -0.0020370 | 0.0045148 | -0.4248301 | 2.3276286 | 1.0697255 | 2.6961902 | 0.1329125 |
| J3 | 0.0080719 | -0.0244734 | 3.8177135 | 4.6942550 | 1.0458528 | 0.9116915 | 0.9044308 |
| J4 | -0.0167661 | 0.0135863 | 0.0514710 | 4.1479845 | 1.4664724 | 2.6583830 | 0.5105733 |
| J5 | -0.0224920 | -0.0065751 | -5.2459234 | 1.0591226 | 1.8493249 | 3.4855856 | 0.4611490 |
| J6 | 0.0024419 | 0.0287763 | -5.6083168 | 5.6414840 | 1.1572256 | 4.6694389 | 2.1517274 |
| J7 | -0.0011204 | 0.0047996 | 0.0092878 | 2.8535327 | 1.0051375 | 0.7036156 | 1.6926768 |
| J8 | 0.0007157 | -0.0022319 | -2.1983833 | 4.0340397 | 0.4485013 | 1.1168611 | 1.2018595 |
| J9 | -0.0018298 | -0.0255393 | -0.0228652 | 0.6146843 | 3.0638726 | 5.6059616 | 2.1866173 |
| J10 | 0.0000257 | 0.0005047 | 0.3201838 | 4.3057878 | 2.7289270 | 3.1826315 | 1.8748329 |
| J11 | -0.0054352 | -0.0039805 | 5.5073683 | 4.9397423 | 2.3809745 | 5.0080307 | 1.4770154 |
| J12 | -0.0051552 | -0.0021995 | -1.1230654 | 0.2654220 | 1.4912686 | 2.0968191 | 0.8169792 |
| J13 | -0.0139684 | -0.0209987 | -0.2058980 | 3.2416578 | 1.5232357 | 6.1723007 | 0.1766462 |
| J14 | 0.0002352 | 0.1892648 | -3.1443034 | 1.5040134 | 2.6306860 | 1.6052789 | 1.7627707 |
| J15 | -0.0000000 | -0.0000000 | 3.1408861 | 2.6156740 | 1.5328672 | 0.8933251 | 1.5751851 |
| J16 | 0.0006897 | -0.0081792 | 0.0020128 | 2.3925079 | 2.8009763 | 6.1531039 | 2.5557602 |
| J17 | -0.0031152 | 0.0136432 | -5.3476986 | 2.1183581 | 0.0638548 | 1.0279100 | 1.5965980 |
| J18 | -0.0017860 | 0.0108283 | 4.7369894 | 3.1738548 | 0.7636444 | 5.3693241 | 0.7452688 |
| J19 | 0.0018374 | 0.0049068 | -3.3205195 | 6.2554242 | 1.4038136 | 2.9239762 | 2.5366105 |
| J20 | -0.0277801 | 0.0091944 | -5.3549641 | 4.4434589 | 1.4551931 | 1.5475781 | 2.6612992 |
| J21 | 0.0041646 | -0.0166041 | -3.3322854 | 0.6846282 | 3.0520374 | 1.7018792 | 2.7893998 |
| J22 | 0.0000009 | 0.0000313 | -3.0495066 | 0.9343671 | 2.7049211 | 0.5199461 | 2.2727047 |
| J23 | -0.0000084 | 0.0010963 | -0.0006658 | 3.1206800 | 1.6070482 | 3.1452084 | 1.1670140 |
| J24 | -0.0000012 | -0.0000799 | -3.1360206 | 4.0982825 | 1.5721129 | 2.5002720 | 1.1525346 |
| J25 | 0.0000002 | -0.0004185 | 0.0031319 | 3.4420810 | 1.7613342 | 1.5697501 | 1.0283187 |
| J26 | 0.0020336 | -0.0073588 | 4.0942521 | 0.9075140 | 0.2446625 | 2.2060874 | 1.6375071 |
| J27 | -0.0000553 | -0.0010036 | 4.9943172 | 5.3283506 | 1.5334776 | 0.9404995 | 0.8395574 |
| J28 | -0.0000000 | 0.0000000 | 0.1121366 | 3.0576468 | 1.4275625 | 4.9022935 | 1.0367738 |
| J29 | 0.0397676 | -0.0731267 | -3.8955221 | 3.9290100 | 1.8607197 | 4.3966298 | 0.2128720 |
| J30 | -0.0043289 | -0.0004889 | 0.0225620 | 0.4551551 | 2.0607770 | 5.2065020 | 0.8515532 |
| J31 | -0.0000000 | -0.0000000 | 1.3097664 | 0.1586296 | 1.7650486 | 1.0551604 | 1.1270299 |
| J32 | -0.0190177 | -0.0228611 | 5.2808310 | 2.3446311 | 2.4232133 | 3.9626008 | 1.2580937 |
| J33 | -0.0013058 | 0.0135149 | -4.8925144 | 1.3683621 | 0.3656969 | 3.9644794 | 1.5340138 |
| J34 | 0.0010230 | 0.0002264 | 1.7576806 | 6.1214536 | 0.2390766 | 4.4378035 | 3.1158965 |
| J35 | 0.0048682 | 0.0095578 | 0.3597662 | 0.3228353 | 1.3869263 | 0.7536865 | 2.6250067 |
| J36 | -0.0000000 | -0.0000000 | 1.5854824 | 3.3191120 | 2.0794952 | 4.0855904 | 1.1710340 |
| J37 | -0.0044644 | -0.0003456 | -0.5747631 | 5.2699083 | 3.0786518 | 1.9947510 | 1.7162821 |
| J38 | 0.0244210 | -0.0038875 | 2.1616213 | 3.6358294 | 1.5153191 | 4.0741130 | 2.7213213 |
| J39 | -0.0000000 | -0.0000000 | 0.9336887 | 0.9851553 | 1.1974226 | 1.5175501 | 2.7189210 |
| J40 | 0.0008087 | -0.0060859 | -2.9891078 | 4.0685474 | 3.0744235 | 4.6173671 | 0.5605312 |
| J41 | 0.0013559 | -0.0329749 | -3.2064616 | 1.2826463 | 0.0881117 | 0.5257150 | 2.8596018 |
| J42 | -0.0007740 | -0.0209728 | 0.0528558 | 1.9938864 | 0.0873607 | 6.2076760 | 2.1297105 |
| J43 | -0.0000000 | -0.0000000 | 3.1475883 | 1.0504189 | 2.4049891 | 3.5383422 | 1.6931246 |
| J44 | -0.0000037 | -0.0009319 | -3.3980705 | 0.1973234 | 0.4073555 | 0.9940381 | 2.0694432 |
| J45 | -0.0000017 | 0.0000011 | 0.0040334 | 3.1488965 | 1.5682867 | 4.7194054 | 2.1801629 |
| J46 | 0.0000088 | 0.0007472 | 0.0028655 | 4.0970651 | 1.5102849 | 4.7155424 | 2.1283157 |
| J47 | 0.0000022 | -0.0003614 | 0.0013670 | 3.4424820 | 1.7675429 | 3.4528521 | 2.0123445 |
| J48 | -0.0001320 | 0.0056488 | 0.2017305 | 5.6793013 | 0.1747458 | 2.1348791 | 1.9167609 |
| J49 | 0.0115267 | 0.0133999 | 1.1881921 | 3.7592402 | 0.4996378 | 0.8157576 | 1.4375100 |
| J50 | 0.0014294 | -0.0109103 | 3.1051807 | 2.4212652 | 1.3924624 | 4.6117448 | 2.3756823 |
| J51 | 0.0002509 | -0.0003312 | -3.3815737 | 5.2559391 | 2.1061002 | 3.7684605 | 2.2619758 |
| J52 | 0.0050997 | 0.0083485 | 5.8352326 | 0.0447103 | 0.7946150 | 4.1030558 | 1.7865101 |
| J53 | -0.0365942 | -0.0075524 | 1.1488931 | 6.0886101 | 2.4734860 | 2.1133928 | 0.6211457 |
| J54 | 0.0013267 | 0.0005511 | -3.9428330 | 0.9512030 | 2.5197298 | 4.1242447 | 1.4629505 |
| J55 | -0.0000000 | 0.0000000 | 3.0109597 | 0.0384079 | 1.7946166 | 4.7180778 | 2.0025006 |
| J56 | -0.0005924 | 0.0031458 | 0.8705622 | 3.5431695 | 1.9271469 | 4.7364962 | 2.3168569 |
| J57 | -0.0001958 | -0.0000627 | -0.3696067 | 4.2521537 | 1.7036767 | 4.5640367 | 0.2452291 |
| J58 | -0.0081955 | -0.0032099 | 0.3416875 | 4.9835965 | 1.3575918 | 0.2764550 | 2.6864917 |
| J59 | -0.0000000 | 0.0000000 | 0.5099307 | 4.0889442 | 0.7946337 | 0.0808687 | 1.5059097 |
| J60 | -0.0082240 | 0.0060116 | 4.4479316 | 5.1925803 | 1.6838010 | 5.9745412 | 0.4675802 |
| J61 | -0.0023312 | -0.0055051 | 0.0504120 | 6.1047587 | 1.9095915 | 2.6731386 | 0.2771198 |
| J62 | 0.0000992 | -0.0030073 | 2.6843213 | 3.2842705 | 0.8817324 | 4.9243183 | 1.6731205 |
| J63 | 0.0000897 | 0.0003362 | -5.2579932 | 4.6521921 | 0.4837780 | 3.0530298 | 1.1447252 |
| J64 | 0.0011533 | 0.0006430 | -4.4769087 | 0.4456515 | 1.1848876 | 1.8495579 | 1.4171600 |
| J65 | -0.0458067 | 0.0459852 | 0.1601105 | 3.8101938 | 1.7044647 | 4.3042041 | 0.5156526 |
| J66 | 0.0079716 | 0.0171494 | 0.2833506 | 0.7355054 | 0.2899416 | 2.2720571 | 1.0065217 |
| J67 | -0.0000000 | 0.0000000 | 0.0571738 | 3.0292282 | 2.7861292 | 1.8647618 | 1.4805117 |
| J68 | -0.0012340 | -0.0024412 | -2.9231912 | 2.3370915 | 0.9431604 | 4.6094944 | 1.9936266 |
| J69 | -0.0000000 | -0.0000000 | 1.5602186 | 1.1480637 | 2.5954809 | 3.2927847 | 1.8672264 |
| J70 | 0.0038343 | 0.0032929 | -3.3738746 | 1.9137118 | 1.5652576 | 4.8400460 | 2.1639053 |
| J71 | -0.0000891 | 0.0000853 | 2.4444757 | 0.7896322 | 1.5713099 | 3.4178340 | 2.7268736 |
| J76 | -0.0000401 | -0.0000274 | 5.1601146 | 3.3181200 | 2.0181057 | 4.7238008 | 1.2419031 |
| J78 | -0.0092863 | -0.0071466 | -1.3355733 | 0.0009452 | 1.5944086 | 0.6071161 | 1.1995332 |
| J79 | -0.0061097 | 0.0055595 | -5.2336958 | 4.1697387 | 0.6221123 | 4.4879503 | 1.3273113 |
| J80 | -0.0000000 | -0.0000000 | 2.5526134 | 6.2409859 | 1.5867991 | 2.2390170 | 1.2448405 |
| J81 | 0.0005473 | -0.0007612 | -1.5756106 | 3.1696606 | 1.5360931 | 4.8155974 | 1.6875127 |
| J82 | -0.0105832 | 0.0051254 | -4.8083313 | 0.0731051 | 3.1185741 | 6.0932539 | 1.7293399 |
| J83 | -0.0009627 | -0.0015376 | -1.2715445 | 3.3528668 | 3.0903465 | 4.3113380 | 1.2882064 |
| J84 | 0.0000886 | -0.0001274 | -3.0927979 | 0.5613720 | 0.9141990 | 5.1730411 | 0.9901529 |
| J85 | 0.0079364 | 0.0268775 | 0.0826045 | 4.8806195 | 1.3516010 | 1.5513808 | 0.3416917 |
| J86 | -0.0077003 | -0.0008668 | 0.0477756 | 3.4742787 | 1.0327427 | 4.7690548 | 0.5747059 |
| J87 | -0.0045786 | 0.0045179 | -0.5525460 | 5.0009466 | 1.7501023 | 5.4763534 | 0.9706826 |
| J88 | 0.0642291 | 0.0027556 | 4.0288329 | 2.3200055 | 1.2894427 | 1.4675617 | 1.3375535 |
| J89 | -0.0025885 | -0.0247849 | -1.9358513 | 0.3406113 | 2.8603994 | 4.8451796 | 0.9897685 |
| J90 | -0.0002215 | -0.0002280 | -0.9107454 | 2.5737338 | 2.5749417 | 1.3889726 | 1.6692917 |
| J91 | -0.0000000 | 0.0000000 | 0.0128004 | 5.0373692 | 2.0701231 | 1.2161610 | 0.8878211 |
| J92 | -0.0092608 | 0.0042913 | 5.9529900 | 3.1797194 | 0.8051523 | 0.5317979 | 0.9054991 |
It is not that easy to visualise a pierced polyhedron in 3D. These interactive drawings are parallel projections produced by a Python script using the parameters x, y, α, θ1, ϕ1, θ2, ϕ2 from the above table. The faces of the hole are invisible at first as those parameters project them to the sides of a polygon. Each polyhedron can then be rotated about the x-axis by dragging up or down with the left-button of the mouse and rotated about the y-axis by dragging left or right with the left-button of the mouse. Clicking on the name brings up an enlarged drawing, in which the polygon can also be rotated about the z-axis using the scroll wheel of the mouse.
| J72 missing | ||
| J73 missing | J74 missing | J75 missing |
| J77 missing | ||
Christopher B. Jones 2023-08-25