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2683:{\displaystyle M={\begin{pmatrix}v_{0}^{T}\\\vdots \\v_{n}^{T}\end{pmatrix}}{\begin{pmatrix}v_{0}&\ldots &v_{n}\end{pmatrix}}={\begin{pmatrix}v_{0}\cdot v_{0}&v_{0}\cdot v_{1}&\cdots &v_{0}\cdot v_{n}\\v_{1}\cdot v_{0}&v_{1}\cdot v_{1}&\cdots &v_{1}\cdot v_{n}\\\vdots &\vdots &\ddots &\vdots \\v_{n}\cdot v_{0}&v_{n}\cdot v_{1}&\cdots &v_{n}\cdot v_{n}\end{pmatrix}},\qquad U={\begin{pmatrix}1&1&\cdots &1\\1&1&\cdots &1\\\vdots &\vdots &\ddots &\vdots \\1&1&\cdots &1\end{pmatrix}}\,.}
1647:{\displaystyle M^{T}M={\begin{pmatrix}v_{0}^{T}&1\\v_{1}^{T}&1\\\vdots &\vdots \\v_{n}^{T}&1\end{pmatrix}}{\begin{pmatrix}v_{0}&v_{1}&\cdots &v_{n}\\1&1&\cdots &1\end{pmatrix}}={\begin{pmatrix}v_{0}\cdot v_{0}+1&v_{0}\cdot v_{1}+1&\cdots &v_{0}\cdot v_{n}+1\\v_{1}\cdot v_{0}+1&v_{1}\cdot v_{1}+1&\cdots &v_{1}\cdot v_{n}+1\\\vdots &\vdots &\ddots &\vdots \\v_{n}\cdot v_{0}+1&v_{n}\cdot v_{1}+1&\cdots &v_{n}\cdot v_{n}+1\end{pmatrix}}}
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rather than the
Frobenius norm that goes in that formula. If by original research you mean that no one has ever discovered and published this before now, ... I very much doubt that. But if you mean that my library isn't big enough that I could find the citation myself ... then I agree! I will let
3555:
that includes the
January 1886 issue of Educational Times and (although it was interesting) I found no reference to Clifford writing about prime confines in that January 1886 issue. My best guess is that the 1886 reference year is a typo and it was never checked by any of the article's editors.
653:
There is a question about the volume formula that is contention. I am hoping that someone has a textbook handy that shows this formula and thus we can add a citation to the article and keep the formula. Specifically, until a few days ago the article has indicated, correctly IMHO, that
295:
I think that a reference to simplices in functional analysis is missing here. Terminologies such as "Choquet simplex" or "Bauer simplex" are often met. As far as I understand, they are infinite-dimensional extensions of the notion of simplex discussed in that article.
3396:. So, how should we modify the article to clarify this seeming inconsistency? As far as I am concerned, anyone with ideas should go ahead and make edits. If we find that there is contention over those edits we can then continue the discussion here. Thanks! —
3575:
The volume is shown as |det...| but shouldn't that be either det or |...| for the short form of the determinant? I didn't know the formula, so I don't want to change it myself, but it looks like a notational error. It is confusing (to me), however.
1731:
Dot products between vectors from the origin to the vertices are not translation-invariant; only dot products between vectors from vertices to other vertices are. It’s quite easy to see that the formula is wrong: for example, it gives an area of
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For example, in 2d you would get (0, 1) and (1, 0), which can not be the correct normal vectors of an equilateral triangle. Further, calculating the angle between two such vectors leads to arccos(1/(n-1)), which is not correct.
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The right hand side depends only upon the dot products, which means that it depends only upon the vector lengths and the angles between the vectors. In particular, it does not depend upon the number of dimensions.
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about the earliest known uses of some mathematical terms, it states that
Clifford wrote about "prime confines" in the January 1886 issue of Educational Times, but yesterday I spent a couple hours scanning through
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This is definitely not original research. The well-known molecular geometry in this example can be found in most any first-year college chemistry textbook and also at
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The article's history section states that
Clifford "wrote about these shapes in 1886" but that was seven years after his death. Perhaps 1886 was when that writing was
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resembling a
Schlegel diagram of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a
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Ha I edited before I saw this comment, so I'm glad you endorsed that :). In the lead section, I don't see any reason to specify the ambient space
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to this talk page. The
Simplex article is a little inconsistent about the dimensionality of the space containing a simplex. Specifically, if a
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vertices; such as the 3 vertices of a triangle (2-simplex) fit well in a 2-dimensional plane. But one of our main examples is placing a
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Both of those issues appear if the normals are instead given by (-n, 1, ..., 1) and permutations, I think that's the correct answer.
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I think this makes sense, the definitions of linear independence and affine independence do not specify dimensions.
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it sit here for a while and hope that someone has a better familiarity with the literature than I. Thank you —
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fails to compute the correct volume then that would help me to understand the objection to the formula. —
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hold true, instead of trying to use a previous paragraph with modifications.01:51, 10 November 2020 (UTC)
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is a general parallelotope, the same assertions hold except that it is no longer true, in dimension : -->
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the
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3107:{\textstyle \mathrm {Volume} ={\tfrac {1}{n!}}{\sqrt {\sum _{i,j}(\operatorname {adj} M)_{ij}}}}
2920:{\textstyle \mathrm {Volume} ={\tfrac {1}{n!}}{\sqrt {\sum _{i,j}(\operatorname {adj} M)_{ij}}}}
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then the question of the truth of the statement is highly relevant. In particular, we say that
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Which article? I don't see any reference to
Choquet simplex or Bauer simplex in the
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2142:{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}{\sqrt {\det \left-\det \left}}\,,}
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is a mapping that preserves distances and angles (and hence preserves volumes) if
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I believe that it is the square root of the sum of the (un-squared) elements of
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has questioned whether this formula applies in a
Euclidean space with more than
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921:{\displaystyle \mathrm {Volume} ={1 \over n!}\left(\det \left\right)^{1/2}\,,}
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It appears that the normal vectors in the dihedral angles sections are wrong:
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of that possibly negative value that is the actual volume, which is what the
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In the first formula, the determinant could come out negative. It is the
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Thanks for pointing that out. At some point I plan to create a dedicated
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gives the same value. The answer is yes, because that expression equals
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https://en.wikipedia.org/Simplex#Dihedral_angles_of_the_regular_n-simplex
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I did the calculations and got the same result as you, so I changed it.—
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420:-hypercube by the linear isomorphism that sends the canonical basis of
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Only a little off topic ... do you know of an algorithm that computes
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et al.: Is it reasonable to add the following volume formula?
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covering all the n-dimensional simplices and related topics. –
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937:-simplex's vertices are in a Euclidean space with more than
3511:. Does this inconsistency extend to the body, as well? --
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If you know of a citation that supports the notability of
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I agree with you now. Thank you for being persistent. —
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bonds with one hydrogen atom and forms a line segment,
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can resemble a simplex if one is to connect each atom.
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but I haven't been able to verify that reference. In
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is symmetric in the vertices and works even when the
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is symmetric in the vertices and works even when the
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be regular, but many interesting simplices are not.
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dimensions, I bring the topic to this discussion. —
101:, a collaborative effort to improve the coverage of
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1686:If there is a counterexample where the formula
979:Especially if this formula is includable under
3346:Dimensionality of space containing the simplex
613:. In particular, the geometry of methane, CH
274:This page has archives. Sections older than
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2986:Right about that formula, corrected above. —
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284:when more than 10 sections are present.
204:Knowledge:WikiProject Uniform Polytopes
154:
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2797:Or mix and match somehow. Thank you —
1016:. Another way of saying that is that
515:I removed this bullet that reads like
207:Template:WikiProject Uniform Polytopes
3308:2601:547:500:E930:944C:F88D:FBB3:E92F
7:
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188:This article is within the scope of
95:This article is within the scope of
3584:2A02:58:157:9B00:70B:DBE4:BC88:CA19
547:bonds with two hydrogen atoms in a
488:{\displaystyle e_{1},\ldots ,e_{n}}
402:contains this incoherent sentence:
38:It is of interest to the following
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3446:{\displaystyle \mathbb {R} ^{17}}
1035:. Another way of saying that is
511:Questionable "Applications" entry
278:may be automatically archived by
115:Knowledge:WikiProject Mathematics
3504:{\displaystyle \mathbb {R} ^{3}}
3475:{\displaystyle \mathbb {R} ^{2}}
503:Much better: State exactly what
442:{\displaystyle \mathbf {R} ^{n}}
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190:WikiProject Uniform Polytopes
109:and see a list of open tasks.
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3535:16:35, 7 February 2023 (UTC)
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1054:and the question is whether
645:Volume formula in more than
500:This is way too confusing.
388:21:19, 13 October 2020 (UTC)
372:09:01, 13 October 2020 (UTC)
336:17:22, 6 December 2019 (UTC)
316:14:52, 6 December 2019 (UTC)
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2821:That’s cute but probably
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2825:. Another cute formula:
141:project's priority scale
3221:15:46, 9 May 2024 (UTC)
627:File:5-simplex verf.png
98:WikiProject Mathematics
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688:
683:
619:Schlegel diagram
596:
591:
494:
492:
491:
486:
484:
483:
465:
464:
448:
446:
445:
440:
438:
437:
432:
283:
267:
239:
231:
212:
211:
208:
205:
202:
185:
178:
177:
172:
160:
153:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
3733:
3732:
3728:
3727:
3726:
3724:
3723:
3722:
3703:
3702:
3692:
3687:
3637:
3612:
3605:
3603:
3573:
3489:
3484:
3483:
3460:
3455:
3454:
3431:
3426:
3425:
3398:
3393:
3389:
3385:
3378:
3374:
3367:
3363:
3359:
3351:
3348:
3323:
3306:, for example.
3282:
3281:
3262:
3261:
3242:
3241:
3235:
3204:
3179:
3139:
3138:
3116:
3089:
3048:
3014:
3013:
2957:
2947:
2902:
2861:
2827:
2826:
2799:
2728:
2695:
2694:
2668:
2667:
2662:
2657:
2652:
2646:
2645:
2640:
2635:
2630:
2624:
2623:
2618:
2613:
2608:
2602:
2601:
2596:
2591:
2586:
2576:
2557:
2556:
2546:
2533:
2531:
2526:
2516:
2503:
2501:
2491:
2478:
2475:
2474:
2469:
2464:
2459:
2453:
2452:
2442:
2429:
2427:
2422:
2412:
2399:
2397:
2387:
2374:
2371:
2370:
2360:
2347:
2345:
2340:
2330:
2317:
2315:
2305:
2292:
2285:
2274:
2273:
2263:
2261:
2256:
2246:
2239:
2231:
2230:
2212:
2211:
2205:
2204:
2182:
2170:
2169:
2161:Alternatively,
2120:
2119:
2109:
2107:
2102:
2092:
2085:
2077:
2076:
2058:
2057:
2051:
2050:
2028:
2026:
2022:
2004:
2003:
1998:
1993:
1987:
1986:
1976:
1974:
1969:
1959:
1952:
1944:
1943:
1938:
1920:
1919:
1914:
1908:
1907:
1902:
1880:
1878:
1874:
1859:
1826:
1825:
1824:The expression
1813:
1811:
1780:
1702:
1687:
1661:
1637:
1636:
1620:
1607:
1605:
1600:
1584:
1571:
1569:
1553:
1540:
1537:
1536:
1531:
1526:
1521:
1515:
1514:
1498:
1485:
1483:
1478:
1462:
1449:
1447:
1431:
1418:
1415:
1414:
1398:
1385:
1383:
1378:
1362:
1349:
1347:
1331:
1318:
1311:
1300:
1299:
1294:
1289:
1284:
1278:
1277:
1267:
1265:
1260:
1250:
1248:
1238:
1231:
1223:
1222:
1217:
1199:
1198:
1193:
1187:
1186:
1181:
1163:
1162:
1157:
1135:
1118:
1113:
1112:
1086:
1076:
1075:and we can use
1064:
1055:
1046:
1036:
1017:
1013:
1009:
991:
984:
957:
944:
882:
881:
876:
871:
866:
860:
859:
849:
847:
842:
832:
830:
820:
813:
805:
804:
799:
781:
780:
775:
769:
768:
763:
745:
744:
739:
717:
715:
711:
707:
703:
702:
692:
659:
658:
651:
616:
594:
589:
513:
475:
456:
451:
450:
427:
422:
421:
396:
344:
293:
279:
268:
262:
244:
209:
206:
203:
200:
199:
166:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
3731:
3729:
3721:
3720:
3715:
3705:
3704:
3701:
3700:
3640:LaundryPizza03
3636:
3633:
3632:
3631:
3600:absolute value
3572:
3569:
3548:this reference
3540:
3539:
3538:
3537:
3498:
3493:
3469:
3464:
3440:
3435:
3377:-simplex into
3347:
3344:
3343:
3342:
3295:
3292:
3289:
3269:
3249:
3234:
3231:
3230:
3229:
3228:
3227:
3226:
3225:
3224:
3223:
3189:
3186:
3182:
3178:
3172:
3169:
3166:
3162:
3157:
3154:
3151:
3147:
3135:
3099:
3096:
3092:
3088:
3085:
3082:
3079:
3074:
3071:
3068:
3064:
3054:
3051:
3047:
3041:
3037:
3034:
3031:
3028:
3025:
3022:
3003:
3002:
3001:
3000:
2999:
2998:
2988:Anders Kaseorg
2979:
2978:
2977:
2976:
2940:
2939:
2929:Anders Kaseorg
2912:
2909:
2905:
2901:
2898:
2895:
2892:
2887:
2884:
2881:
2877:
2867:
2864:
2860:
2854:
2850:
2847:
2844:
2841:
2838:
2835:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2734:
2731:
2727:
2722:
2718:
2715:
2712:
2709:
2706:
2703:
2691:
2690:
2679:
2672:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2647:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2625:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2603:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2581:
2579:
2574:
2571:
2566:
2561:
2553:
2549:
2545:
2540:
2536:
2532:
2530:
2527:
2523:
2519:
2515:
2510:
2506:
2502:
2498:
2494:
2490:
2485:
2481:
2477:
2476:
2473:
2470:
2468:
2465:
2463:
2460:
2458:
2455:
2454:
2449:
2445:
2441:
2436:
2432:
2428:
2426:
2423:
2419:
2415:
2411:
2406:
2402:
2398:
2394:
2390:
2386:
2381:
2377:
2373:
2372:
2367:
2363:
2359:
2354:
2350:
2346:
2344:
2341:
2337:
2333:
2329:
2324:
2320:
2316:
2312:
2308:
2304:
2299:
2295:
2291:
2290:
2288:
2283:
2278:
2270:
2266:
2262:
2260:
2257:
2253:
2249:
2245:
2244:
2242:
2235:
2227:
2222:
2218:
2214:
2213:
2210:
2207:
2206:
2201:
2196:
2192:
2188:
2187:
2185:
2180:
2177:
2138:
2130:
2124:
2116:
2112:
2108:
2106:
2103:
2099:
2095:
2091:
2090:
2088:
2081:
2073:
2068:
2064:
2060:
2059:
2056:
2053:
2052:
2047:
2042:
2038:
2034:
2033:
2031:
2025:
2021:
2018:
2014:
2008:
2002:
1999:
1997:
1994:
1992:
1989:
1988:
1983:
1979:
1975:
1973:
1970:
1966:
1962:
1958:
1957:
1955:
1948:
1942:
1939:
1935:
1930:
1926:
1922:
1921:
1918:
1915:
1913:
1910:
1909:
1906:
1903:
1899:
1894:
1890:
1886:
1885:
1883:
1877:
1873:
1865:
1862:
1858:
1853:
1849:
1846:
1843:
1840:
1837:
1834:
1816:Anders Kaseorg
1810:
1807:
1806:
1805:
1804:
1803:
1802:
1801:
1800:
1799:
1769:
1768:
1767:
1766:
1765:
1764:
1754:Anders Kaseorg
1744:), and (0, 1,
1724:
1723:
1722:
1721:
1681:
1680:
1656:
1655:
1654:
1641:
1635:
1632:
1627:
1623:
1619:
1614:
1610:
1606:
1604:
1601:
1599:
1596:
1591:
1587:
1583:
1578:
1574:
1570:
1568:
1565:
1560:
1556:
1552:
1547:
1543:
1539:
1538:
1535:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1516:
1513:
1510:
1505:
1501:
1497:
1492:
1488:
1484:
1482:
1479:
1477:
1474:
1469:
1465:
1461:
1456:
1452:
1448:
1446:
1443:
1438:
1434:
1430:
1425:
1421:
1417:
1416:
1413:
1410:
1405:
1401:
1397:
1392:
1388:
1384:
1382:
1379:
1377:
1374:
1369:
1365:
1361:
1356:
1352:
1348:
1346:
1343:
1338:
1334:
1330:
1325:
1321:
1317:
1316:
1314:
1309:
1304:
1298:
1295:
1293:
1290:
1288:
1285:
1283:
1280:
1279:
1274:
1270:
1266:
1264:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1236:
1234:
1227:
1221:
1218:
1214:
1209:
1205:
1201:
1200:
1197:
1194:
1192:
1189:
1188:
1185:
1182:
1178:
1173:
1169:
1165:
1164:
1161:
1158:
1154:
1149:
1145:
1141:
1140:
1138:
1133:
1130:
1125:
1121:
1106:
1105:
947:Anders Kaseorg
929:
928:
917:
910:
906:
902:
897:
892:
886:
880:
877:
875:
872:
870:
867:
865:
862:
861:
856:
852:
848:
846:
843:
839:
835:
831:
827:
823:
819:
818:
816:
809:
803:
800:
796:
791:
787:
783:
782:
779:
776:
774:
771:
770:
767:
764:
760:
755:
751:
747:
746:
743:
740:
736:
731:
727:
723:
722:
720:
714:
710:
706:
698:
695:
691:
686:
682:
679:
676:
673:
670:
667:
650:
643:
642:
641:
614:
573:
572:
512:
509:
482:
478:
474:
471:
468:
463:
459:
436:
431:
395:
392:
391:
390:
343:
340:
339:
338:
292:
289:
286:
285:
273:
270:
269:
264:
260:
258:
255:
254:
246:
245:
240:
234:
228:
224:
223:
220:
219:
216:
215:
213:
186:
174:
173:
161:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
3730:
3719:
3716:
3714:
3711:
3710:
3708:
3699:
3695:
3690:
3684:
3683:
3680:
3677:
3672:
3668:
3667:
3666:
3665:
3661:
3657:
3653:
3649:
3645:
3641:
3634:
3630:
3626:
3622:
3618:
3615:
3601:
3597:
3596:
3595:
3593:
3589:
3585:
3581:
3570:
3568:
3567:
3563:
3559:
3554:
3549:
3545:
3536:
3532:
3528:
3524:
3523:
3522:
3518:
3514:
3496:
3467:
3438:
3423:
3419:
3418:
3417:
3416:
3412:
3408:
3404:
3401:
3381:
3370:
3355:
3345:
3341:
3337:
3333:
3329:
3326:
3320:
3319:
3318:
3317:
3313:
3309:
3293:
3290:
3287:
3267:
3247:
3239:
3232:
3222:
3218:
3214:
3210:
3207:
3187:
3184:
3176:
3155:
3152:
3149:
3145:
3136:
3134:
3130:
3126:
3122:
3119:
3097:
3094:
3086:
3083:
3080:
3072:
3069:
3066:
3062:
3052:
3049:
3045:
3039:
3011:
3010:
3009:
3008:
3007:
3006:
3005:
3004:
2997:
2993:
2989:
2985:
2984:
2983:
2982:
2981:
2980:
2975:
2971:
2967:
2963:
2960:
2953:
2951:
2944:
2943:
2942:
2941:
2938:
2934:
2930:
2910:
2907:
2899:
2896:
2893:
2885:
2882:
2879:
2875:
2865:
2862:
2858:
2852:
2824:
2820:
2819:
2818:
2817:
2813:
2809:
2805:
2802:
2794:
2792:
2788:
2767:
2758:
2752:
2749:
2746:
2732:
2729:
2725:
2720:
2677:
2670:
2664:
2659:
2654:
2649:
2642:
2637:
2632:
2627:
2620:
2615:
2610:
2605:
2598:
2593:
2588:
2583:
2577:
2572:
2569:
2564:
2559:
2551:
2547:
2543:
2538:
2534:
2528:
2521:
2517:
2513:
2508:
2504:
2496:
2492:
2488:
2483:
2479:
2471:
2466:
2461:
2456:
2447:
2443:
2439:
2434:
2430:
2424:
2417:
2413:
2409:
2404:
2400:
2392:
2388:
2384:
2379:
2375:
2365:
2361:
2357:
2352:
2348:
2342:
2335:
2331:
2327:
2322:
2318:
2310:
2306:
2302:
2297:
2293:
2286:
2281:
2276:
2268:
2264:
2258:
2251:
2247:
2240:
2233:
2225:
2220:
2216:
2208:
2199:
2194:
2190:
2183:
2178:
2175:
2168:
2167:
2166:
2162:
2158:
2156:
2152:
2136:
2128:
2122:
2114:
2110:
2104:
2097:
2093:
2086:
2079:
2071:
2066:
2062:
2054:
2045:
2040:
2036:
2029:
2023:
2016:
2012:
2006:
2000:
1995:
1990:
1981:
1977:
1971:
1964:
1960:
1953:
1946:
1940:
1933:
1928:
1924:
1916:
1911:
1904:
1897:
1892:
1888:
1881:
1875:
1863:
1860:
1856:
1851:
1821:
1817:
1808:
1798:
1794:
1790:
1786:
1783:
1777:
1776:
1775:
1774:
1773:
1772:
1771:
1770:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1735:
1730:
1729:
1728:
1727:
1726:
1725:
1720:
1716:
1712:
1708:
1705:
1698:
1695:
1691:
1685:
1684:
1683:
1682:
1679:
1675:
1671:
1667:
1664:
1657:
1639:
1633:
1630:
1625:
1621:
1617:
1612:
1608:
1602:
1597:
1594:
1589:
1585:
1581:
1576:
1572:
1566:
1563:
1558:
1554:
1550:
1545:
1541:
1533:
1528:
1523:
1518:
1511:
1508:
1503:
1499:
1495:
1490:
1486:
1480:
1475:
1472:
1467:
1463:
1459:
1454:
1450:
1444:
1441:
1436:
1432:
1428:
1423:
1419:
1411:
1408:
1403:
1399:
1395:
1390:
1386:
1380:
1375:
1372:
1367:
1363:
1359:
1354:
1350:
1344:
1341:
1336:
1332:
1328:
1323:
1319:
1312:
1307:
1302:
1296:
1291:
1286:
1281:
1272:
1268:
1262:
1255:
1251:
1243:
1239:
1232:
1225:
1219:
1212:
1207:
1203:
1195:
1190:
1183:
1176:
1171:
1167:
1159:
1152:
1147:
1143:
1136:
1131:
1128:
1123:
1119:
1111:
1110:
1108:
1107:
1102:
1099:
1095:
1092:
1089:
1082:
1079:
1073:
1070:
1067:
1061:
1058:
1052:
1049:
1042:
1039:
1033:
1030:
1027:
1023:
1020:
1006:
1002:
998:
994:
987:
982:
978:
977:
976:
975:
971:
967:
963:
960:
954:
948:
942:
938:
934:
915:
908:
904:
900:
895:
890:
884:
878:
873:
868:
863:
854:
850:
844:
837:
833:
825:
821:
814:
807:
801:
794:
789:
785:
777:
772:
765:
758:
753:
749:
741:
734:
729:
725:
718:
712:
704:
696:
693:
689:
684:
657:
656:
655:
648:
644:
640:
636:
632:
628:
624:
620:
612:
608:
604:
603:
602:
601:
597:
592:
586:
585:
582:
579:
570:
566:
562:
558:
554:
550:
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3578:— Preceding
3574:
3553:this archive
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40:WikiProjects
1778:Thank you —
943:. Because
565:a structure
557:tetrahedron
112:Mathematics
103:mathematics
59:Mathematics
3707:Categories
3544:published
3527:Dukeleimao
3422:in general
3354:Dukeleimao
1740:), (1, 0,
1008:for every
941:dimensions
649:dimensions
3644:Polytopes
3635:irregular
3394:(0, 0, 1)
3390:(0, 1, 0)
3386:(1, 0, 0)
3238:Quantling
631:Anita5192
623:5-simplex
525:chemistry
380:Anita5192
328:Anita5192
326:article.—
3656:—Tamfang
3625:contribs
3617:uantling
3606:det(...)
3580:unsigned
3411:contribs
3403:uantling
3336:contribs
3328:uantling
3217:contribs
3209:uantling
3129:contribs
3121:uantling
2970:contribs
2962:uantling
2812:contribs
2804:uantling
1793:contribs
1785:uantling
1715:contribs
1707:uantling
1674:contribs
1666:uantling
970:contribs
962:uantling
553:nitrogen
541:fluorine
360:unsigned
312:contribs
300:unsigned
276:365 days
242:Archives
195:inactive
169:inactive
3676:Laundry
1692:!) det
988:: R → R
981:WP:CALC
621:of the
609:. See
578:Laundry
569:halogen
537:a point
529:p-block
304:Gamesou
139:on the
30:B-class
3453:as in
3392:, and
563:forms
561:carbon
559:, and
545:oxygen
36:scale.
3679:Pizza
3233:Signs
2823:WP:OR
2693:Then
581:Pizza
571:atom.
3660:talk
3621:talk
3588:talk
3562:talk
3531:talk
3517:talk
3482:and
3407:talk
3332:talk
3312:talk
3213:talk
3125:talk
2992:talk
2966:talk
2948:adj
2933:talk
2808:talk
2165:Let
1789:talk
1758:talk
1711:talk
1670:talk
1060:(LM)
1057:(LM)
1012:and
966:talk
635:talk
549:bent
533:Neon
505:does
384:talk
368:talk
332:talk
308:talk
131:High
3652:can
3646:to
3558:Gj7
3513:JBL
3081:adj
2927:. —
2894:adj
2762:det
2741:det
2020:det
1872:det
1752:. —
1732:½√(
1688:(1/
1083:= I
1043:= I
709:det
625:in
523:In
449:to
407:If
3709::
3696:)
3693:c̄
3682:03
3662:)
3627:)
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3590:)
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3413:)
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3382:+1
3371:+1
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3248:∓
3240::
3219:)
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2994:)
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2935:)
2897:
2876:∑
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2638:⋱
2633:⋮
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1329:⋅
1292:⋯
1263:⋯
1196:⋮
1191:⋮
1096:=
1094:LM
1072:LM
1032:Lv
1024:=
1005:Lv
1003:·
1001:Lu
999:=
995:·
972:)
968:|
874:⋯
845:⋯
778:⋮
773:⋮
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629:.—
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470:…
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310:•
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3375:k
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3330:(
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3177:M
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