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PENTATOPE
NUMBER
Pentatope is the name of a specific geometric
figure that human beings cannot directly visualize because it does
not exist as a 3-dimensional object. The suffix "-tope" refers to
the "cells" which comprise geometric figures that exist in a greater
number of dimensions than the three in which human beings physically
reside. Humans can partially visualize various polytopes by
examining the idealized, mathematically derived, "geometric shadow"
such higher dimensional figures would cast upon a 2-D surface when
illuminated from behind. A 3-D "tetra-hedron" is composed of four
equilateral triangles whose planes provide its "four faces. " A
"penta-tope" contains "five cells" in the form of five Tetrahedrons
enclosed within a Hypersphere.
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If we take five tetrahedrons and merge them
according to their bilateral symmetry we form the 3D shadow of
a pentatope, the simplest 4D figure. Each of its five
dimensional faces is a regular
tetrahedron. |
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If we think of these five tetrahedrons and
their merge symmetry as two loops of five plane faces, then
the folding and unfolding of the pentatope through 3D space
forms a Moebius strip. This creates the pentatope's self
referencing duality. The shadow of this self-referencing in 2D
is a pentagram inscribed in a pentagon.
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Pentatope Numbers (4-tetrahedron numbers) can be found in
the fifth column of Pascal`s triangle, which are of course the
sum of the tetrahedral numbers.The pentatope numbers are:
1,5,15,35,70,126,210,330,495,715,1001,1365,1820,2380,3060,
3876,4845,5985,7315,8855,10626,12650,14950,17550,20475,23751,
27405,31465,35960,40920,46376,52360,58905,66045,73815,82251,
91390,101270,111930,123410,135751,148995,163185,178365...
Binomial coefficients binomial (n,4).
Number of intersection points of diagonals of convex
n-gon.
Also the number of equilateral triangles with vertices
in a equilateral triangular array of points with n rows
(offset 1), with any orientation.
Start from cubane and attach amino acids according to
the reaction scheme that describes the reaction between the
active sites.
For n>0 a(n)=(-1/8)*coefficient of x in Zagier's
polynomial P_(2n,n) (Zagier's polynomials are used by
pari-gp for acceleration of alternating or positive series)
Figurate numbers based on the 4-dimensional regular
convex polytope called the regular 4-simplex, pentachoron,
5-cell, pentatope or 4-hypertetrahedron with Schlafli symbol
{3,3,3}.
The pentatope numbers can be found in the
fifth column of Pascal`s triangle, starting at row 4 as shown
in the diagram. The first tetrahedral number is 1, the second
is 5, the third is 15, the forth is 35, and so on.
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The sums of the consecutive tetrahedral numbers
(starting from 1) are the pentatope numbers.
For example : 4845 =
1 + 4 + 10 + 20 + 35 + 56 + 84 + 120 +165+ 225 + 286 + 364 + 455 +
560 + 680 + 816 + 969 or the seventeenth pentatope number (17 sums):
Numbers that are
Added |
The Sum
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1 |
1 |
1 + 4 |
5 |
1 + 4 + 10 |
15 |
1 + 4 + 10 + 20 |
35 |
1 + 4 + 10 + 20 + 35 |
70 |
1 + 4 + 10 + 20 + 35 + 56 |
126 |
1 + 4 + 10 + 20 + 35 + 56 + 84 |
210 |
1 + 4 + 10 + 20 + 35 + 56 + 84 + 120
|
330 |
1 + 4 + 10 + 20 + 35 + 56 + 84 + 120 + 165
|
495 |
1 + 4 + 10 + 20 + 35 + 56 + 84 + 120 + 165 +
220 |
715 |
The sum of these ... the new pattern shown in the
right column ... is referenced as the sequence of pentatope numbers.
The pentatope number is a figurate number of the form:
where n= 4, 5,6, ...
As it is known, the tetrahedral number is a
figurate number of the following form:
where n= 3, 4, 5, 6 ....
There is a connection between pentatope numbers
and tetrahedral numbers :
Now, the operator`s relation is obtained:
or:
An infinite sum of the tetrahedral numbers as the
negative pentatope number can be calculated, as shown in the table
below :
Infinite series of the tetrahedral numbers
|
The Sum
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1+4+10+20+35+56+84+120+... |
- 0 |
4+10+20+35+56+84+120+165+... |
- 1 |
10+20+35+56+84+120+165+220+... |
- 5 |
20+35+56+84+120+165+220+286+... |
- 15 |
35+56+84+120+165+220+286+364+... |
- 35 |
56+84+120+165+220+286+364+455+... |
- 70 |
84+120+165+220+286+364+455+560+... |
-126 |
120+165+220+286+364+455+560+680+... |
-210 |
165+220+286+364+455+560+680+816+... |
-330 |
220+286+364+455+560+680+816+969+... |
-495 |
An interesting number`s triangle can be generated
by adding up tetrahedral numbers :
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01 |
01 |
01 |
01 |
....... |
4* 01 |
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04 |
04 |
04 |
04 |
04 |
....... |
4* 05 |
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10 |
10 |
10 |
10 |
10 |
10 |
....... |
4* 15 |
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20 |
20 |
20 |
20 |
20 |
20 |
20 |
....... |
4* 35 |
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35 |
35 |
35 |
35 |
35 |
35 |
35 |
35 |
....... |
4* 70 |
|
56 |
56 |
56 |
56 |
56 |
56 |
56 |
56 |
56 |
....... |
4*126 |
| 84 |
84 |
84 |
84 |
84 |
84 |
84 |
84 |
84 |
84 |
....... |
4*210 |
Also the next number`s triangle can be generated by
adding up pentatope and tetrahedral numbers :
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01 |
01 |
01 |
01 |
01 |
....... |
5* 01 |
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01 |
04 |
04 |
04 |
04 |
04 |
04 |
....... |
5* 05 |
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05 |
10 |
10 |
10 |
10 |
10 |
10 |
10 |
....... |
5* 15 |
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15 |
20 |
20 |
20 |
20 |
20 |
20 |
20 |
20 |
....... |
5* 35 |
|
35 |
35 |
35 |
35 |
35 |
35 |
35 |
35 |
35 |
35 |
....... |
5* 70 |
| 70 |
56 |
56 |
56 |
56 |
56 |
56 |
56 |
56 |
56 |
56 |
....... |
5*126 |
Pentatope numbers satisfy the recurrence relation,
as shown in the table below:
|
point |
line |
triangle |
pyramid |
pentatope |
pentatope |
|
1 |
1 |
1 |
1 |
1 |
5 |
|
1 |
2 |
3 |
4 |
5 |
15 |
|
1 |
3 |
6 |
10 |
15 |
35 |
|
1 |
4 |
10 |
20 |
35 |
70 |
|
1 |
5 |
15 |
35 |
70 |
126 |
|
1 |
6 |
21 |
56 |
126 |
210 |
|
1 |
7 |
28 |
84 |
210 |
330 |
Figurate Numbers: 1, 6, 21, 56, 126,252 . . . , can
be found in the sixth diagonal of Pascal`s triangle, which are of
course the sum of the pentatope numbers.
See Also: Triangular
Number , Tetrahedral
Number
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