Truncated Tetrahedral Numbers

The number associated with my diurnal age today, 28325, has a property that allows it membership of OEIS A100156:

 A100156: structured truncated tetrahedral numbers.

So what is a truncated tetrahedron. It belongs to a class of three-dimensional figurate numbers. They are geometrically constructed by taking a regular tetrahedral number of a given size and removing smaller tetrahedral corners from each of its four vertices.

What does it look like? Figure 1 shows an example.

Figure 1: source

Figure 2 shows the associated net made up from four hexagons and four equilateral triangles.

Figure 2

The details are:

Faces: 8

Edges: 18

Vertices: 12 

A structured truncated tetrahedral number (OEIS A100156) is a three-dimensional figurate number that counts the total number of points (or spheres) in a truncated tetrahedron built outward in nested shells from a single corner vertex, where every polygonal face is populated as a classical vertex-grown figurate polygon rather than a centered polygon.

The initial members of the sequence are: 1, 12, 44, 108, 215, 376, 602, 904, 1293, 1780, 2376, 3092, 3939, 4928, 6070, 7376, 8857, 10524, 12388, 14460, 16751, 19272, 22034, 25048, 28325, 31876, 35712, 39844

 

Figure 3

To quote from Gemini:

The shape first emerged prominently in the 15th and 16th centuries as artists sought to calculate and render perfect three-dimensional forms on flat surfaces:

Leonardo da Vinci & Luca Pacioli (1509): The most celebrated early appearance of the truncated tetrahedron is in Fra Luca Pacioli's treatise, De Divina Proportione. Leonardo da Vinci provided the illustrations, drawing the solid in two revolutionary forms: solidus (a solid form showing surface planes) and vacuus (a skeletal wireframe showing its interior framework). See Figure 3.

Albrecht Dürer (1525): In his foundational work Underweysung der Messung (Instruction in Measurement), German master Albrecht Dürer revolutionized geometric study by publishing the first foldable paper nets. His flat layout of the truncated tetrahedron allowed readers to cut out and construct physical models, making complex perspective easier to perceive in physical space.

Tetraprimes

While I have looked at numbers with four prime factors counted with multiplicity whose reversals also have this property, I've not actually looked at reversible tetraprimes. Numbers with four distinct prime factors are called tetraprimes. The first example of such a number is 1518 where we have:

• \(1518 = 2 \times 3 \times 11 \times 23 \)
• \( 8151 = 3 \times 11 \times 13 \times 19 \)

There are 273 such numbers in the range up to 40000 (permalink):

1518, 2046, 2226, 2262, 2418, 2478, 2618, 2622, 2814, 2838, 2886, 3135, 3927, 4170, 4182, 4386, 4389, 4746, 4785, 4935, 5313, 5394, 5406, 5478, 5565, 5655, 5838, 5874, 6018, 6045, 6222, 6402, 6438, 6474, 6486, 6690, 6699, 6834, 6846, 6882, 7293, 7458, 8106, 8142, 8151, 8162, 8346, 8382, 8385, 8547, 8742, 8745, 9834, 9966, 10434, 10506, 11022, 11346, 11814, 11946, 12243, 12441, 12738, 12765, 13026, 13299, 13542, 13629, 13695, 14105, 14118, 14421, 14469, 14574, 15114, 15873, 16005, 16107, 16359, 16665, 16786, 16962, 16995, 17017, 17358, 17589, 17655, 17754, 17922, 18183, 18258, 18447, 18462, 18546, 18615, 18879, 19434, 19437, 19446, 19578, 19662, 20022, 20055, 20085, 20118, 20145, 20163, 20190, 20262, 20310, 20355, 20382, 20405, 20526, 20553, 20559, 20562, 20658, 20746, 20769, 20774, 20878, 20922, 20958, 22002, 22011, 22074, 22098, 22154, 22242, 22290, 22458, 22515, 22533, 22578, 22695, 22710, 22737, 22755, 22854, 22902, 22946, 22962, 22971, 24115, 24123, 24178, 24186, 24198, 24222, 24270, 24297, 24339, 24465, 24530, 24531, 24582, 24585, 24618, 24645, 24662, 24717, 24766, 24846, 24890, 24906, 24915, 24969, 26013, 26103, 26106, 26187, 26202, 26286, 26358, 26394, 26445, 26481, 26490, 26502, 26535, 26554, 26634, 26642, 26691, 26733, 26895, 26922, 26961, 26978, 26985, 26994, 28083, 28086, 28194, 28245, 28302, 28329, 28410, 28446, 28455, 28497, 28509, 28542, 28545, 28578, 28623, 28626, 28635, 28785, 28866, 28886, 28905, 28974, 28985, 28990, 30162, 30426, 31038, 31062, 31146, 31494, 31515, 31614, 31955, 32142, 32574, 32637, 32682, 32745, 32835, 32865, 33198, 33285, 33522, 33594, 33654, 33666, 33762, 34017, 34221, 34386, 34419, 34518, 34554, 34626, 34854, 35502, 35673, 35706, 35826, 36102, 36195, 36218, 36645, 36966, 36993, 37037, 37074, 37158, 37365, 37605, 37653, 37785, 37851, 38055, 38082, 38181, 38445, 39045, 39666, 39738, 39963

I've also considered biprimes as rectangles and triprimes as rectangular prisms but what about tetraprimes. I asked Gemini the following:

A triprime with prime factors of \(a\), \(b\) and \(c\) can be conceived of as a rectangular prism with sides \(a\), \(b\), \(c\), volume \(a \, b \, c\) and surface area equal to \(2 \times (ab + bc + ac) \). Are there equivalent formulae for volume and surface area of a four dimensional object derived from a tetraprime with four prime factors.

Here was Gemini's response:

The direct geometrical equivalent of a tetraprime with four prime factors \(a\), \(b\), \(c\), and \(d\) is a four-dimensional rectangular orthotope (commonly called a hyperrectangle).

Because a 4D object possesses boundaries in multiple lower dimensions (3D, 2D, and 1D), the equivalent to your 3D formulas expands into a hierarchy of measurements. Just as a 3D prism is bounded by 2D faces, a 4D orthotope is bounded by 3D cells, 2D faces, and 1D edges.

Here are the equivalent geometric formulas derived from a tetraprime:

• 4D Hypervolume (The equivalent of 3D Volume): The total 4D spatial content of the object is simply the product of all four prime factors, which is exactly equal to the tetraprime itself.$$V_4 = a \cdot b \cdot c \cdot d$$

• 3D Surface Volume (The structural equivalent of 3D Surface Area): In 3D geometry, "surface area" measures the (n-1)-dimensional boundary of the object. For a 4D object, its outer boundary is three-dimensional. A hyperrectangle is enclosed by 8 solid 3D rectangular prisms (called "cells"). There are two cells for each possible triplet of the dimensions.$$S_3 = 2(abc + abd + acd + bcd)$$

• 2D Total Face Area (The literal equivalent of 2D Area): If you are looking specifically for flat, two-dimensional surface area, a hyperrectangle has 24 flat 2D rectangular faces. For every pair of dimensions, there are 4 identical faces throughout the 4D structure.$$A_2 = 4(ab + ac + ad + bc + bd + cd)$$

• 1D Total Edge Length: For complete geometric breakdown, a hyperrectangle has 32 one-dimensional edges. There are 8 parallel edges for each of the four prime factors.$$L_1 = 8(a + b + c + d)$$

By substituting your four prime factors into \(V_4\) and \(S_3\), you achieve the exact mathematical analogies for volume and surface area as you step up from three dimensions into four.

I'll investigate this further in a subsequent blog.