1616:
cube with a single point. But the 4-simplex is also the join of a triangle and a line segment, and you could consider the shape formed by the join of a square with a line segment as a generalization of a pyramid. This shape has as its facets two square pyramids (the joins of the square with the endpoints of the line segment) and four tetrahedra (the joins of the sides of the square with the whole line segment). Its
477:. As such, the approximation would continually jump from negative to positive to negative again, etc.. The value of x that would seem to work is such that Tan(x)=2x. I know this sounds silly, and probably because it is, but I was awfully bored at the time :P. Thank you very much for your insight, I'll try and get a fairly accurate numerical approximation :)
677:. To follow the naming convention for inverse trig/hyperbolic function, the sought value would be atanc(2). I've only seen the name tanc used on MathWorld, and the name atanc was made up on the spot here, but both really should be standard since equations similar to tan(x)=x are so common in physics. Maybe if I implement both in
1506:) does not, only two of the dimensions (the ones defining the base) are exchangeable. So you could either make the next dimension a second "height" or a third "base" dimension. The resulting two figures would not be the same. If you made specific the type of analog you were considering, it would help.
1937:
Ok, the semester exam is worth half as much as the quarter grades, so you count each quarter grade twice. That means you've effectively got 9 marks to count, plus the final exam, makes the 10 you divide by. The reason your average is less than the individual marks is because it's assuming you get 0%
2007:
So we can deduce that the final exam has a weight of 10% in the final average. If you scoe x% in the final exame then your final average will be 85 + (x/10) %. So you can be certain of a final average of at least 85%, but even if you ace the final exam you can't get a final average better than 95%.
1829:
But a square-based pyramid is an irregular shape - there are any number of irregular 4 dimensional spaces which have pyramidal hyper-faces. It's kinda like if you were a 2D being - you could ask for 3D analogs of things like squares, circles and equilateral triangles - and get answers like "cube",
1823:
A simpler way to explain why there is an ambiguity is that we don't know the form of the extra dimension. In the case of a regular tetrahedron, it's reasonable to assume that the 4th dimensional analog is also a regular figure - so you get the tetrahedron that shrinks to a dot as the 4 dimensional
1615:
are placed but for the combinatorics of what faces the join has it doesn't matter. A three-dimensional pyramid over some base is the join of that base (2-dimensional) with a single point (0-dimensional). The 4-simplex and pyramid-over-cube you're talking about are joins of a regular tetrahedron or
413:
Heyo, I am trying to solve Tan(x)=2x for x, but I have never encountered a function such as this before. The domain is {0<x<Pi/2}. My current efforts in solving it have given me an approximate value of x=1.1655612, but an exact value would be wonderful. Mathematica can't solve it, either, it
1520:
As
Baccyak says, there are multiple things that could be called a 4-D pyramid. I would go with a cube going to a point (a square pyramid is a square going to a point, add one dimension to a square, you get a cube). I think that might be what Baccyak means by adding a third base dimension. I'm not
706:
Is the flux of the vector field F=(xz)i+(x)j+(y)k across the surface of the hemisphere of radius 5 oriented in the direction of the positive y-axis equal to Zero? I'm getting up to a string of double-integrals between 0 and pi, but each term has a sin function that results in everything ending up
441:
works nicely if you take a starting value between 1 and 1.5. Starting further away, say between 0.7 and 0.9, Newton's method sometimes converges to other roots, sometimes cycles, and sometimes does not seem to converge at all. Perhaps someone might be interested in imaging the
414:
would seem. I'm only a high school student, so it would be perfectly reasonable to assume that such a solution is beyond my capabilities (for now), so can anyone help me out, please? I assure you that this is not homework, but rather a curiousity I am following. Many thanks
377:
1742:
AAML is a pyramid whose base is a square pyramid. Its "faces" are four tetrahedra, and two square pyramids (including the base). In fact, once the shape is made, there's no way to identify which of the two square pyramids was the "original"
108:
Recently in
Calculus II, we have started doing some integrals of functions defined by parametric equations (as well as in polar coordinates), my question is how does one prove that area under a parametric curve is given by the formula.
1476:
as it passed through our three-dimensional space, but what's the four-dimensional equivalent of a square-based
Egyptian-type pyramid? Would its cross section(if it fell "face-first") be a cube, a triangular prism, or something else?
464:
It's interesting you mention Newton's method. I was attempting find a root for Sin(x) between -Pi/2 and Pi/2, which is clearly zero, using Newton's method. I was investigating, however, the possibly of taking an initial value of
1938:
on the final exam (which apparently has the same weight as the semester exam). If you want just your average for the bits you've taken so far, not including the final exam at all, just divide by 9 instead of 10. --
1406:
1620:
can be formed as a square pyramid with a single vertex inside it connected by edges to all five pyramid corners. And, as Black Carrot suggests, it is also the join of a square pyramid and a single point.
1902:
The key is that you’re dividing by 10 at the end, and so the original sum of (1) to (5) must be 1000. Assuming each quarter is out of 100, that’s how I deduced that the semester exam is probably out of
1898:
My best guess would be that the semester exam is out of 200 points, not 100: but has equal weighting as the quarters. Thus to get equal weighting he gave them all a value of 200 points by doubling each
1757:
MMAL is a prism with a triangular prism for its base. Its faces include has 4 triangular prisms, in two opposite pairs, and three cubes. Either of the pairs of triangular prisms could count as the base.
491:
Silly, perhaps, but in a good way :). I agree that this problem is intriguing. Note that
Mathematica will be very happy to provide a numerical approximation if you ask it nicely - The following command:
185:
1786:
Thanks, that's very informative! Are there any images, nets, or animations of those shapes? (Most of the 4D geometry pages I've seen just have the perfectly symmetrical ones and their stellations.)
66:
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977:
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25:
85:
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the
1449:
Actually, that was my reading too. Fortunately, the flux through the flat side of the hemisphere is obviously zero, so you get the same answer.
1750:
Any of the last three could perhaps count as the "equivalent" of the square pyramid, but I'd vote for the last. Then, there are the prisms..
1492:
I am not 100% sure about this, but I suspect there is some ambiguity about what the analog actually would be. The tetrahedron's point group
1435:
Ah -- I had misread "surface of the hemisphere" as if it were "half of the surface of a sphere", which would be an unclosed surface. Eric.
722:
1739:
AMAL is a pyramid whose base is a triangular prism. Its "faces" has two tetrahedra, three square pyramids, and the base, a triangular prism.
1793:
1478:
37:
1499:
has triply degenerate symmetries, meaning that the three dimensions are in some sense equivalent or at least exchangeable. The pyramid (
1835:
1436:
1354:
437:
for this solution - if there was, Mathematica would probably have found it. So a numerical approximation is the best that you will get.
395:
1642:
Starting with a shape S, there are many ways to increase its dimension. One way is to add a point in the next dimension. This gives a
1880:
What's the logic behind the doubling and dividing. Also how do I go from 90's (as in my example) to 85? Thanks...I'm really lost.
21:
1376:
615:
Well, tan is actually composed of (complex) exponentials, so one might be able to use the
Lambert W Function for that purpose.
114:
394:
I suppose that works, but I really want to see a constructionist type proof (much like the
Fundamental theorem of Calculus).
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1671:. Another way is to move the shape in the direction of the new dimension, and see what it carves out. This gives you a
629:
I think I've thought about that once and concluded that it is not actually possible. Too lazy to look at it again. --
1830:"sphere" and "tetrahedron". But demanding the 3D analog of an irregular 2D figure would be an unanswerable question.
1417:
1626:
718:
634:
606:
514:
86:
17:
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which was invented for just this purpose. You could just as well invent a function that gives the solution to
1839:
1607:
as placed together in the larger dimensional space. For the geometry of the resulting shape it matters where
1440:
1358:
399:
1022:
805:
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MAML is a prism with pyramids at the top and bottom. The sides consist of 1 cube, and 4 triangular prisms.
1540:
1413:
714:
434:
888:
372:{\displaystyle \int _{\alpha }^{\beta }g(t)\cdot f'(t)\cdot dt=\int _{f(\alpha )}^{f(\beta )}y\cdot dx.}
193:
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1700:. Now we can think of what you get if you apply A and M successively, starting with a line segment
1622:
1511:
630:
602:
559:
510:
451:
383:
1959:
1927:
1887:
1754:
MAAL is a prism with a tetrahedral base. Its faces are 4 triangular prisms, and the 2 tetrahedra.
1746:
AMML is a pyramid with a cube for its base. Its "faces" are 6 square pyramids and the cubic base.
1370:
563:
620:
526:
1979:
1617:
650:
482:
438:
419:
189:
1943:
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1526:
1877:
For example Q1=93%, Q2=90%, SE=93%, Q3=100%, and Q4=94%; my final average would be 85%.
2009:
1907:
1854:
My professor said to figure my final average for the year, without the final exam, to:
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1507:
1125:
447:
443:
379:
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1276:
1188:
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1105:
1002:
982:
962:
785:
765:
745:
1970:
1953:
1921:
1881:
686:
616:
1874:
Then I am to add these numbers together and divide by 10 to get my final average.
674:
1974:
1596:
1469:
646:
478:
415:
1939:
1906:(If everything were out of 100, then the maximum possible would be 900/10=90)
1808:
1522:
1473:
1764:
523:
Perhaps it is worth mentioning that if the equation you wanted to solve was
601:, but it seems no-one has deemed it important enough to give it a name. --
74:
1920:
Sorry, that's percentage not points. I should have been more specific.
1555:
Re adding a second height dimension: in general, the join of two shapes
1643:
1468:
I know that the four-dimensional equivalent of the tetrahedron is the
733:
Yes, it is zero. No messy double integrals are necessary, by the way.
678:
682:
1672:
1521:
quite sure what adding a 2nd height dimension would mean... --
509:
gives 1.1655611852072113068339179779585606691345388476931. --
1401:{\displaystyle \nabla \cdot {\overrightarrow {\mathbf {F} }}}
645:
Your
Mathematica-Fu is very impressive. Thank you very much!
429:
Well, there ic clearly only one solution in the range 0 <
203:
f is the x-coordinate and g is the y-coordinate. Substitute
79:
Welcome to the
Knowledge Mathematics Reference Desk Archives
1704:. (Or, you could start with a single point if you prefer)
1807:
That was a great answer - worth three cents, at least! --
180:{\displaystyle A=\pm \int _{\alpha }^{\beta }g(t)f'(t)dt}
762:
be your surface, which is symmetric under negation of
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1339:
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117:
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1145:have the same flux, which must be zero. Flipping
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1682:If S is a triangle, you get a triangular prism.
8:
1656:If S is a square, you get a square pyramid.
1650:If S is a line segment, you get a triangle.
1870:Take my fourth quarter grade and double it
1861:Take my second quarter grade and double it
1653:If S is a triangle, you get a tetrahedron.
1535:A square pyramid going to a point, maybe.
673:Someone has invented such a function: the
1867:Take my third quarter grade and double it
1858:Take my first quarter grade and double it
1679:If S is a line segment, you get a square.
1571:dimensions) is formed by taking disjoint
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571:
528:
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271:
266:
260:
208:
136:
131:
116:
1732:Your question is about the next step...
707:zero, but it seems an unlikely answer.
409:Solution for Tan(x)=2x. {0<x<Pi/2}
49:
36:
65:
1095:{\displaystyle (x(-z))i+0j+(y)(-k)=-G}
878:{\displaystyle G(x,y,z)=(xz)i+0j+(y)k}
433:< π/2, but I doubt that there is a
43:
1688:If S is a circle, you get a cylinder.
7:
1675:, whose base is the original shape.
1646:, whose base is the original shape.
499:N - 2 x, {x, 1}, AccuracyGoal -: -->
952:{\displaystyle H(x,y,z)=0i+(x)j+0k}
1579:-dimensional affine subspaces of (
1408:is anti-symmetric about the plane
1380:
32:
1969:This type of average is called a
1824:shape translated through 3-space.
1685:If S is a square, you get a cube.
1659:If S is a circle, you get a cone.
1369:I choose the lazy approach. The
1266:{\displaystyle 0(-i)+(-x)j+0k=-H}
1587:+ 1)-dimensional space, placing
1474:tetrahedron dwindling to a point
1389:
1373:is very useful here. Note that
248:{\displaystyle dx=f'(t)\cdot dt}
104:Integrating parametric equations
594:{\displaystyle \tan x=\alpha x}
1239:
1230:
1224:
1215:
1080:
1071:
1068:
1062:
1044:
1041:
1032:
1026:
934:
928:
913:
895:
869:
863:
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812:
349:
343:
335:
329:
306:
300:
286:
280:
233:
227:
168:
162:
151:
145:
1:
1715:Then, in three dimensions...
685:, the world will follow... -
33:
1595:within them, and taking the
1426:Yeah, that was my approach.
1313:is the sum of the fluxes of
1692:Let's call this operation,
1663:Let's call this operation,
1273:, so similarly the flux of
500:55, WorkingPrecision -: -->
2042:
1353:, so must be zero. Eric.
1722:MAL is a triangular prism
1472:, which would resemble a
551:{\displaystyle \log x=2x}
2018:08:49, 23 May 2008 (UTC)
1985:23:30, 22 May 2008 (UTC)
1965:23:11, 22 May 2008 (UTC)
1952:Ahhh. Gotcha. Thanks!
1948:22:56, 22 May 2008 (UTC)
1933:22:46, 22 May 2008 (UTC)
1916:22:40, 22 May 2008 (UTC)
1893:22:34, 22 May 2008 (UTC)
1850:Figuring "Final Average"
1844:15:43, 24 May 2008 (UTC)
1817:20:15, 23 May 2008 (UTC)
1802:00:44, 24 May 2008 (UTC)
1781:02:10, 23 May 2008 (UTC)
1631:15:47, 22 May 2008 (UTC)
1545:15:45, 22 May 2008 (UTC)
1531:15:30, 22 May 2008 (UTC)
1516:15:25, 22 May 2008 (UTC)
1487:15:00, 22 May 2008 (UTC)
1464:Four-dimensional pyramid
1454:11:03, 23 May 2008 (UTC)
1445:10:23, 23 May 2008 (UTC)
1431:21:19, 22 May 2008 (UTC)
1422:20:49, 22 May 2008 (UTC)
1363:12:23, 22 May 2008 (UTC)
738:12:09, 22 May 2008 (UTC)
727:11:39, 22 May 2008 (UTC)
690:10:56, 23 May 2008 (UTC)
655:11:36, 22 May 2008 (UTC)
639:00:14, 23 May 2008 (UTC)
625:21:22, 22 May 2008 (UTC)
611:11:33, 22 May 2008 (UTC)
519:11:20, 22 May 2008 (UTC)
487:10:39, 22 May 2008 (UTC)
456:10:26, 22 May 2008 (UTC)
424:09:02, 22 May 2008 (UTC)
404:23:22, 22 May 2008 (UTC)
388:08:20, 22 May 2008 (UTC)
198:06:23, 22 May 2008 (UTC)
18:Knowledge:Reference desk
1975:Confusing Manifestation
1725:AML is a square pyramid
1293:is 0. But the flux of
1763:MMML is the good ole'
1402:
1347:
1327:
1307:
1287:
1267:
1199:
1179:
1159:
1139:
1116:
1096:
1013:
993:
973:
953:
879:
796:
776:
756:
595:
552:
435:closed-form expression
373:
249:
181:
87:current reference desk
1864:Take my semester exam
1736:AAAL is a pentachoron
1403:
1348:
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1719:AAL is a tetrahedron
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702:Flux across surface.
570:
527:
446:for this equation ?
259:
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115:
560:elementary function
353:
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141:
1398:
1371:divergence theorem
1343:
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1138:{\displaystyle -G}
1135:
1112:
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989:
969:
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591:
564:Lambert W function
548:
369:
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177:
127:
1983:
1804:
1792:comment added by
1771:Hope that helps!
1708:AL is a triangle,
1696:oving the shape,
1575:-dimensional and
1396:
1346:{\displaystyle H}
1326:{\displaystyle G}
1306:{\displaystyle F}
1286:{\displaystyle H}
1205:plane, we obtain
1198:{\displaystyle z}
1178:{\displaystyle y}
1158:{\displaystyle H}
1115:{\displaystyle G}
1019:plane, we obtain
1012:{\displaystyle y}
992:{\displaystyle x}
972:{\displaystyle G}
795:{\displaystyle y}
775:{\displaystyle x}
755:{\displaystyle S}
729:
713:comment added by
687:Fredrik Johansson
558:, there is a non-
473:value would be -x
93:
92:
73:
72:
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1977:
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1618:Schlegel diagram
1563:dimensions) and
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38:Mathematics desk
34:
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1667:dding a point,
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1414:Prestidigitator
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715:Damian Eldridge
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439:Newton's method
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1794:69.111.191.122
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1747:
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1711:ML is a square
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1623:David Eppstein
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1479:69.111.191.122
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1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1008:
988:
968:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
791:
771:
751:
740:
703:
700:
699:
698:
697:
696:
695:
694:
693:
692:
664:
663:
662:
661:
660:
659:
658:
657:
643:
642:
641:
631:Meni Rosenfeld
603:Meni Rosenfeld
590:
587:
584:
581:
578:
575:
547:
544:
541:
538:
535:
532:
521:
511:Meni Rosenfeld
498:
497:
496:
495:
494:
493:
492:
474:
470:
469:in which the x
466:
459:
458:
444:Newton fractal
410:
407:
392:
391:
368:
365:
362:
359:
356:
351:
348:
345:
342:
337:
334:
331:
328:
324:
320:
317:
314:
311:
308:
305:
302:
298:
295:
291:
288:
285:
282:
279:
274:
269:
265:
244:
241:
238:
235:
232:
229:
225:
222:
218:
215:
212:
176:
173:
170:
167:
164:
160:
157:
153:
150:
147:
144:
139:
134:
130:
126:
123:
120:
105:
102:
100:
97:
95:
91:
90:
82:
81:
71:
70:
64:
48:
41:
40:
31:
15:
14:
13:
10:
9:
6:
4:
3:
2:
2038:
2019:
2015:
2011:
2006:
2005:
2004:
2003:
2002:
2001:
2000:
1999:
1998:
1997:
1986:
1981:
1976:
1972:
1971:weighted mean
1968:
1967:
1966:
1963:
1957:
1951:
1950:
1949:
1945:
1941:
1936:
1935:
1934:
1931:
1925:
1919:
1918:
1917:
1913:
1909:
1905:
1901:
1897:
1896:
1895:
1894:
1891:
1885:
1878:
1875:
1869:
1866:
1863:
1860:
1857:
1856:
1855:
1849:
1845:
1841:
1837:
1836:70.116.10.189
1834:
1833:
1828:
1827:
1822:
1821:
1818:
1814:
1810:
1806:
1803:
1799:
1795:
1791:
1785:
1784:
1783:
1782:
1778:
1774:
1766:
1762:
1759:
1756:
1753:
1752:
1751:
1745:
1741:
1738:
1735:
1734:
1733:
1728:MML is a cube
1727:
1724:
1721:
1718:
1717:
1716:
1710:
1707:
1706:
1705:
1703:
1699:
1695:
1687:
1684:
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1678:
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1674:
1670:
1666:
1658:
1655:
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1649:
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1628:
1624:
1619:
1614:
1610:
1606:
1602:
1598:
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1578:
1574:
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1566:
1562:
1558:
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1534:
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1528:
1524:
1519:
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1513:
1509:
1505:
1498:
1491:
1490:
1489:
1488:
1484:
1480:
1475:
1471:
1463:
1455:
1452:
1448:
1447:
1446:
1442:
1438:
1437:217.42.199.10
1434:
1432:
1429:
1425:
1424:
1423:
1419:
1415:
1411:
1393:
1383:
1372:
1368:
1367:
1364:
1360:
1356:
1355:144.32.89.104
1340:
1320:
1300:
1280:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1236:
1233:
1227:
1221:
1218:
1212:
1192:
1172:
1152:
1132:
1129:
1109:
1089:
1086:
1083:
1077:
1074:
1065:
1059:
1056:
1053:
1050:
1047:
1038:
1035:
1029:
1006:
986:
966:
946:
943:
940:
937:
931:
925:
922:
919:
916:
910:
907:
904:
901:
898:
892:
872:
866:
860:
857:
854:
851:
848:
842:
839:
833:
827:
824:
821:
818:
815:
809:
789:
769:
749:
741:
739:
736:
732:
731:
730:
728:
724:
720:
716:
712:
701:
691:
688:
684:
680:
676:
675:tanc function
672:
671:
670:
669:
668:
667:
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665:
656:
652:
648:
644:
640:
636:
632:
628:
627:
626:
622:
618:
614:
613:
612:
608:
604:
588:
585:
582:
579:
576:
573:
565:
561:
545:
542:
539:
536:
533:
530:
522:
520:
516:
512:
508:
507:
506:
505:
504:
503:
490:
489:
488:
484:
480:
463:
462:
461:
460:
457:
453:
449:
445:
440:
436:
432:
428:
427:
426:
425:
421:
417:
408:
406:
405:
401:
397:
396:69.54.143.177
389:
385:
381:
366:
363:
360:
357:
354:
346:
340:
332:
326:
322:
318:
315:
312:
309:
303:
296:
293:
289:
283:
277:
272:
267:
263:
242:
239:
236:
230:
223:
220:
216:
213:
210:
202:
201:
200:
199:
195:
191:
187:
174:
171:
165:
158:
155:
148:
142:
137:
132:
128:
124:
121:
118:
110:
103:
98:
96:
88:
84:
83:
80:
77:
76:
68:
61:
57:
53:
47:
42:
39:
35:
27:
23:
19:
1961:¡Talk to me!
1929:¡Talk to me!
1889:¡Talk to me!
1879:
1876:
1873:
1853:
1770:
1749:
1731:
1714:
1701:
1697:
1693:
1691:
1668:
1664:
1662:
1641:
1637:
1612:
1608:
1604:
1600:
1592:
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1556:
1537:Black Carrot
1500:
1493:
1467:
1409:
959:. Flipping
705:
430:
412:
393:
188:
111:
107:
94:
78:
1788:—Preceding
1597:convex hull
1470:pentachoron
1165:across the
979:across the
709:—Preceding
190:A math-wiki
26:Mathematics
1451:Algebraist
1428:Algebraist
735:Algebraist
2010:Gandalf61
1908:GromXXVII
1773:mike40033
1765:tesseract
1508:Baccyak4H
501:65], 50]
448:Gandalf61
380:Bo Jacoby
50:<<
1899:quarter.
1790:unsigned
1639:My 2c..
723:contribs
711:unsigned
24: |
22:Archives
20: |
1980:Say hi!
1644:pyramid
1412:=0. --
802:. Let
617:Paxinum
562:called
255:to get
89:pages.
1102:; so
679:mpmath
647:Vvitor
479:Vvitor
416:Vvitor
99:May 22
67:May 23
46:May 21
1940:Tango
1809:Tango
1743:base.
1673:prism
1523:Tango
683:sympy
69:: -->
63:: -->
62:: -->
44:<
16:<
2014:talk
1955:§hep
1944:talk
1923:§hep
1912:talk
1903:200.
1883:§hep
1840:talk
1813:talk
1798:talk
1777:talk
1627:talk
1611:and
1603:and
1591:and
1567:(in
1559:(in
1541:talk
1527:talk
1512:Yak!
1483:talk
1441:talk
1418:talk
1359:talk
1333:and
1122:and
885:and
742:Let
719:talk
681:and
651:talk
635:talk
621:talk
607:talk
515:talk
483:talk
452:talk
420:talk
400:talk
384:talk
194:talk
1958:•
1926:•
1886:•
1599:of
782:or
574:tan
531:log
60:Jun
56:May
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1973:.
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1914:)
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1800:)
1779:)
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1443:)
1420:)
1394:→
1384:⋅
1381:∇
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1258:−
1234:−
1219:−
1130:−
1087:−
1075:−
1036:−
725:)
721:•
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623:)
609:)
586:α
577:
534:
517:)
485:)
454:)
422:)
402:)
386:)
358:⋅
347:β
333:α
323:∫
310:⋅
290:⋅
273:β
268:α
264:∫
237:⋅
196:)
138:β
133:α
129:∫
125:±
58:|
54:|
2012:(
1982:)
1978:(
1942:(
1910:(
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1767:.
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1665:A
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1255:=
1252:k
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1237:x
1231:(
1228:+
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1222:i
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1185:-
1173:y
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1133:G
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1090:G
1084:=
1081:)
1078:k
1072:(
1069:)
1066:y
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1060:+
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1054:0
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1030:x
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1007:y
999:-
987:x
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911:z
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864:(
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825:,
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816:x
813:(
810:G
790:y
770:x
750:S
717:(
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633:(
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589:x
583:=
580:x
546:x
543:2
540:=
537:x
513:(
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475:1
471:2
467:1
465:x
450:(
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418:(
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382:(
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364:x
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344:(
341:f
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330:(
327:f
319:=
316:t
313:d
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304:t
301:(
297:′
294:f
287:)
284:t
281:(
278:g
243:t
240:d
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231:t
228:(
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217:=
214:x
211:d
192:(
175:t
172:d
169:)
166:t
163:(
159:′
156:f
152:)
149:t
146:(
143:g
122:=
119:A
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