Tetrahedral Decomposition

tetra_decomp.py

#
# Following https://micronote.tech/2020/12/Generating-STL-Models-with-Python/
#
# Remember Python indexes starting at 0!
# Coordinates derived from R script tetrahedral_decomp.R
# The .stl file created contains one unit of the 4 from a tetrahedral decomposition (they
# are all identical of course).  Dimensions default to mm, so the object is about 1.3 mm
# in its longest dimension.  For 3D printing we scaled by 2500% to get an object about
# an inch tall, which is a good size.

import numpy as np
from stl import mesh

vertices = np.array([
    [0.0, 0.0, 0.0],
    [1.0, 1.0, 1.0],
    [1.0, 0.0, 0.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, 1.0],
    [0.333, 0.333, -0.333],
    [0.333, -0.333, 0.333],
    [-0.333, 0.333, 0.333],
])

faces = np.array([
  [4, 1, 7],
  [7, 1, 3],

  [4, 6, 1],
  [6, 2, 1],

  [6, 0, 5],
  [6, 5, 2],

  [0, 7, 3],
  [0, 3, 5],

  [4, 7, 0],
  [4, 0, 6],

  [1, 2, 5],
  [1, 5, 3],
])

shape = mesh.Mesh(np.zeros(faces.shape[0], dtype=mesh.Mesh.dtype))
for i, f in enumerate(faces):
    for j in range(3):
        shape.vectors[i][j] = vertices[f[j], :]

shape.save("tetra_decomp.stl")

Tetrahedral_Decomposition.R

# Decomposition of a Tetrahedron
# Bryan Hanson, May 2024
# Inspired by https://twitter.com/dreugeniacheng/status/1787725859605942480

library("rgl")

# corners of tetrahedron inscribed inside of a cube centered on 0, 0, 0 in Cartesian coordinates
verts <- data.frame( 
	x = c(1, 1, -1, -1),
	y = c(1, -1, 1, -1),
	z = c(1, -1, -1, 1))
rownames(verts) <- c("A", "B", "C", "D") # label the vertices

# need center of each face (ABC, ABD, ACD, BCD)
abc <- colMeans(verts[c("A", "B", "C"),])
abd <- colMeans(verts[c("A", "B", "D"),])
acd <- colMeans(verts[c("A", "C", "D"),])
bcd <- colMeans(verts[c("B", "C", "D"),])

# need mid-point of each edge (6 edges total, no unique order exists)
edges <- data.frame(
	x = c(1, 1, 1, -1, -1, -1, -1, 1, -1, 1, 1, -1),
	y = c(1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1),
	z = c(1, -1, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1))
rownames(edges) <- c("A1", "B1", "B2", "C1", "C2", "D1",
                     "D2", "A2", "C3", "A3", "B3", "D3")

# need mid-point of AB, BC, CD, DA, CA, BD (pairs of rows in edges data frame)
ab <- colMeans(edges[1:2, ])
bc <- colMeans(edges[3:4, ])
cd <- colMeans(edges[5:6, ])
da <- colMeans(edges[7:8, ])
ca <- colMeans(edges[9:10, ])
bd <- colMeans(edges[11:12, ])

# plot it; no attempt at polishing the color scheme(s)
points3d(verts, size = 15, point_antialias = TRUE) # vertices

points3d(abc, size = 15, point_antialias = TRUE, col = "red") # face centers
points3d(abd, size = 15, point_antialias = TRUE, col = "red")
points3d(acd, size = 15, point_antialias = TRUE, col = "red")
points3d(bcd, size = 15, point_antialias = TRUE, col = "red")

segments3d(edges$x, edges$y, edges$z, line_antialias = TRUE, col = "blue", lwd = 4) # edges

points3d(ab, size = 15, point_antialias = TRUE, col = "green") # edge mid-points
points3d(bc, size = 15, point_antialias = TRUE, col = "green")
points3d(cd, size = 15, point_antialias = TRUE, col = "green")
points3d(da, size = 15, point_antialias = TRUE, col = "green")
points3d(ca, size = 15, point_antialias = TRUE, col = "green")
points3d(bd, size = 15, point_antialias = TRUE, col = "green")

points3d(c(0, 0, 0), size = 15, point_antialias = TRUE) # the center

# add the vertices of one piece
points3d(c(0, 0, 0), size = 20, point_antialias = TRUE, col = "orange") # the center
points3d(verts[1,], size = 20, point_antialias = TRUE, col = "orange") # vertex A
points3d(ab, size = 20, point_antialias = TRUE, col = "orange") # mid-point ab
points3d(ca, size = 20, point_antialias = TRUE, col = "orange") # mid-point ca
points3d(da, size = 20, point_antialias = TRUE, col = "orange") # mid-point da
points3d(abc, size = 20, point_antialias = TRUE, col = "orange") # face center abc
points3d(abd, size = 20, point_antialias = TRUE, col = "orange") # face center abd
points3d(acd, size = 20, point_antialias = TRUE, col = "orange") # face center acd

# coordinates of a single piece
piece <- rbind(c(0, 0, 0), verts[1, ], ab, ca, da, abc, abd, acd)
rownames(piece) <- 1:8

# new plot (close any open rgl windows)
# show the piece vertices marked inside the tetrahedron
points3d(verts, size = 15, point_antialias = TRUE) # vertices

points3d(abc, size = 15, point_antialias = TRUE) # face centers
points3d(abd, size = 15, point_antialias = TRUE)
points3d(acd, size = 15, point_antialias = TRUE)
points3d(bcd, size = 15, point_antialias = TRUE)

segments3d(edges$x, edges$y, edges$z, line_antialias = TRUE, col = "blue", lwd = 4) # edges

points3d(ab, size = 15, point_antialias = TRUE, col = "red") # edge mid-points
points3d(bc, size = 15, point_antialias = TRUE, col = "red")
points3d(cd, size = 15, point_antialias = TRUE, col = "red")
points3d(da, size = 15, point_antialias = TRUE, col = "red")
points3d(ca, size = 15, point_antialias = TRUE, col = "red")
points3d(bd, size = 15, point_antialias = TRUE, col = "red")

points3d(piece, size = 20, point_antialias = TRUE, col = "green")

text3d(piece, texts = as.character(1:8), cex = 5) # label for working out the stl details

Use tetrahedral_decomposition.R to generate coordinates, and then use tetra_decomp.py to generate the .stl file.