The number associated with my diurnal age today, 28325, has a property that allows it membership of OEIS A100156:
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.
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Figure 1: source |
Figure 2 shows the associated net made up from four hexagons and four equilateral triangles.
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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
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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.
