## The Rupert property of the Johnson Solids

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.

namexyαθ1ϕ1θ2ϕ2
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