# The compound of five tetrahedra

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# The compound of five tetrahedra

In this previous post I was wrong when I said I didn’t have a tool to compute the intersection of the compound of five tetrahedra:

This can be achieved with the help of the **rcdd** package in R.

This package is a wrapper of the C library **cddlib**. It implements the *double description* of convex polyhedra:

the

*V-description*is the description by the vertices of the polyhedron;the

*H-description*is the description by a set of linear inequalities.

Once we get the H-description of two or more polyhedra, it is straightforward to get the H-description of their intersection: it suffices to join all the linear inequalities. Then, **cddlib** will give us the corresponding V-representation. That’s it.

The **cddlib** library is also usable in Python, with the **pycddlib** library. I’m going to show how to compute the intersection of the five tetrahedra forming the compound with R and with Python.

## The R way

Here are the vertices of the five tetrahedra:

# the twenty vertices #### phi <- (1 + sqrt(5)) / 2 a <- 1 / sqrt(3) b <- a / phi c <- a * phi vertices <- rbind( c( a, a, a), c( a, a, -a), c( a, -a, a), c(-a, -a, a), c(-a, a, -a), c(-a, a, a), c( 0, b, -c), c( 0, -b, -c), c( 0, -b, c), c( c, 0, -b), c(-c, 0, -b), c(-c, 0, b), c( b, c, 0), c( b, -c, 0), c(-b, -c, 0), c(-b, c, 0), c( 0, b, c), c( a, -a, -a), c( c, 0, b), c(-a, -a, -a) ) # the five tetrahedra #### th1 <- vertices[c(17L, 14L, 2L, 11L), ] th2 <- vertices[c(18L, 1L, 4L, 5L), ] th3 <- vertices[c(19L, 6L, 15L, 7L), ] th4 <- vertices[c( 3L, 13L, 12L, 8L), ] th5 <- vertices[c(20L, 16L, 10L, 9L), ]

To make a V-description with **rcdd**, one uses the function `makeV`

:

library(rcdd) V1 <- makeV(th1) V2 <- makeV(th2) V3 <- makeV(th3) V4 <- makeV(th4) V5 <- makeV(th5)

Let’s look at a V-description in R:

V1 ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0 1 0.0000000 0.3568221 0.9341724 ## [2,] 0 1 0.3568221 -0.9341724 0.0000000 ## [3,] 0 1 0.5773503 0.5773503 -0.5773503 ## [4,] 0 1 -0.9341724 0.0000000 -0.3568221 ## attr(,"representation") ## [1] "V"

Each row starts with a zero and a one, followed by the coordinates of the vertex. I don’t remember what the zero means. The one means that the subsequent numbers represent a vertex (it is also possible to encode rays and lines in a V-description).

Now we get the H-representations with the help of the `scdd`

function:

H1 <- scdd(V1) H2 <- scdd(V2) H3 <- scdd(V3) H4 <- scdd(V4) H5 <- scdd(V5)

Let’s look at a H-description:

H1 ## $output ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0 1 -2.802517e+00 -2.220446e-16 -1.070466e+00 ## [2,] 0 1 -8.881784e-16 1.070466e+00 2.802517e+00 ## [3,] 0 1 1.070466e+00 -2.802517e+00 9.614813e-17 ## [4,] 0 1 1.732051e+00 1.732051e+00 -1.732051e+00 ## attr(,"representation") ## [1] "H"

In the first column, a zero indicates that the corresponding rows represents an inequality. The number in the second column is the right hand side of the inequality, and the subsequent three numbers are the coefficients of the linear inequality.

Now, let’s join all the linear inequalities. That is, we stack the five matrices:

H <- rbind(H1$output, H2$output, H3$output, H4$output, H5$output)

The `scdd`

function converts from the V-representation to the H-representation and vice-versa. Hence we get the vertices of the intersection by applying this function to the above H-representation:

(V <- scdd(H) ) ## $output ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0 1 -2.205282e-01 -8.212094e-16 3.568221e-01 ## [2,] 0 1 4.240675e-16 -3.568221e-01 2.205282e-01 ## [3,] 0 1 5.654233e-16 3.568221e-01 2.205282e-01 ## [4,] 0 1 2.205282e-01 0.000000e+00 3.568221e-01 ## [5,] 0 1 3.568221e-01 -2.205282e-01 4.574370e-17 ## [6,] 0 1 3.568221e-01 2.205282e-01 -4.574370e-17 ## [7,] 0 1 2.205282e-01 3.202059e-16 -3.568221e-01 ## [8,] 0 1 1.017762e-15 3.568221e-01 -2.205282e-01 ## [9,] 0 1 6.785079e-16 -3.568221e-01 -2.205282e-01 ## [10,] 0 1 -2.205282e-01 -5.010035e-16 -3.568221e-01 ## [11,] 0 1 -3.568221e-01 2.205282e-01 2.795606e-16 ## [12,] 0 1 -3.568221e-01 -2.205282e-01 -9.718761e-17 ## attr(,"representation") ## [1] "V"

The vertices are given in the last three columns. Let’s extract them:

vertices <- V$output[, c(3L, 4L, 5L)]

There are twelve vertices. The intersection is a regular icosahedron. Now we will plot it. First, we compute the convex hull of these vertices with the **cxhull** package. The icosahedron is convex, so its convex hull is itself. But **cxhull** also provides the faces and the edges that are used for plotting.

library(cxhull) icosahedron <- cxhull(vertices, triangulate = TRUE)

There’s a convenient function in **cxhull** to plot a convec hull with **rgl**, namely `plotConvexHull3d`

:

library(rgl) open3d(windowRect = c(50, 50, 562, 562)) view3d(10, 80, zoom = 0.7) plotConvexHull3d( icosahedron, palette = hcl.colors(256, "BuPu"), bias = 0.25, edgesColor = "yellow", tubesRadius = 0.06, spheresRadius = 0.08 )

## The Python way

Here is the Python code computing the intersection of the five tetrahedra:

import numpy as np import cdd as cdd # the twenty vertices phi = (1 + np.sqrt(5)) / 2 a = 1 / np.sqrt(3) b = a / phi c = a * phi vertices = np.vstack( ( np.array([a, a, a]), np.array([a, a, -a]), np.array([a, -a, a]), np.array([-a, -a, a]), np.array([-a, a, -a]), np.array([-a, a, a]), np.array([0, b, -c]), np.array([0, -b, -c]), np.array([0, -b, c]), np.array([c, 0, -b]), np.array([-c, 0, -b]), np.array([-c, 0, b]), np.array([b, c, 0]), np.array([b, -c, 0]), np.array([-b, -c, 0]), np.array([-b, c, 0]), np.array([0, b, c]), np.array([a, -a, -a]), np.array([c, 0, b]), np.array([-a, -a, -a]), ) ) # tetrahedra vertices tetra1Idxs = [16, 13, 1, 10] tetra2Idxs = [17, 0, 3, 4] tetra3Idxs = [18, 5, 14, 6] tetra4Idxs = [2, 12, 11, 7] tetra5Idxs = [19, 15, 9, 8] th1 = vertices[tetra1Idxs, :] th2 = vertices[tetra2Idxs, :] th3 = vertices[tetra3Idxs, :] th4 = vertices[tetra4Idxs, :] th5 = vertices[tetra5Idxs, :] # make the V-representation of each tetrahedron; you have to prepend the # vertices array with a column of ones v = np.column_stack((np.ones(4), th1)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly1 = cdd.Polyhedron(mat) v = np.column_stack((np.ones(4), th2)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly2 = cdd.Polyhedron(mat) v = np.column_stack((np.ones(4), th3)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly3 = cdd.Polyhedron(mat) v = np.column_stack((np.ones(4), th4)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly4 = cdd.Polyhedron(mat) v = np.column_stack((np.ones(4), th5)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly5 = cdd.Polyhedron(mat) # H-representations of the tetrahedra h1 = poly1.get_inequalities() h2 = poly2.get_inequalities() h3 = poly3.get_inequalities() h4 = poly4.get_inequalities() h5 = poly5.get_inequalities() # join the five sets of linear inequalities; this will give the intersection hintersection = np.vstack((h1, h2, h3, h4, h5)) # make the V-representation of the intersection mat = cdd.Matrix(hintersection, number_type='float') mat.rep_type = cdd.RepType.INEQUALITY polyintersection = cdd.Polyhedron(mat) # get the vertices; they are given in a matrix prepended by a column of ones vintersection = polyintersection.get_generators() # get rid of the column of ones n_inequalities = vintersection.row_size intersection = np.array([ vintersection[i][1:4] for i in range(n_inequalities) ])

Results are the same as the ones we got with R:

array([[-2.20528179e-01, -8.21209393e-16, 3.56822090e-01], [ 4.24067460e-16, -3.56822090e-01, 2.20528179e-01], [ 5.65423280e-16, 3.56822090e-01, 2.20528179e-01], [ 2.20528179e-01, 0.00000000e+00, 3.56822090e-01], [ 3.56822090e-01, -2.20528179e-01, 4.57437043e-17], [ 3.56822090e-01, 2.20528179e-01, -4.57437043e-17], [ 2.20528179e-01, 3.20205930e-16, -3.56822090e-01], [ 1.01776190e-15, 3.56822090e-01, -2.20528179e-01], [ 6.78507937e-16, -3.56822090e-01, -2.20528179e-01], [-2.20528179e-01, -5.01003463e-16, -3.56822090e-01], [-3.56822090e-01, 2.20528179e-01, 2.79560644e-16], [-3.56822090e-01, -2.20528179e-01, -9.71876138e-17]])

It is possible to get the edges of the polyhedron with **pycddlib**; see how I did in this post. And to get the convex hull with its faces, one can use `scipy.spatial.ConvexHull(intersection)`

.

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