Boerdijk-Coxeter helix (tetrahelix)

Octahelix.R

library(rgl)

# tetrahedra vertices
r <- 3*sqrt(3)/10
theta <- acos(-2/3)
h <- 1/sqrt(10)
n <- 20
i <- 1:n
p <- cbind(r*cos(i*theta), r*sin(i*theta), i*h)

# the octahedron within a tetrahedron, given the tetrahedron vertices
# this function returns a mesh
octaFromTetra <- function(vs) {
  p1 <- (vs[, 1] + vs[, 2]) / 2
  p2 <- (vs[, 1] + vs[, 3]) / 2
  p3 <- (vs[, 1] + vs[, 4]) / 2
  p4 <- (vs[, 2] + vs[, 3]) / 2
  p5 <- (vs[, 2] + vs[, 4]) / 2
  p6 <- (vs[, 3] + vs[, 4]) / 2
  P <- rbind(p1, p2, p3, p4, p5, p6)
  faces <- cbind(
    c(1L, 2L, 3L),
    c(1L, 2L, 4L),
    c(1L, 3L, 5L),
    c(1L, 4L, 5L),
    c(6L, 2L, 3L),
    c(6L, 2L, 4L),
    c(6L, 3L, 5L),
    c(6L, 4L, 5L)
  )
  tmesh3d(
    vertices = t(P),
    indices  = faces
  )
}

# take the tetrahedron faces
th0 <- tetrahedron3d()$it

# tetrahedra vertices - there are n-3 tetrahedra in the helix
VS <- lapply(1:(n-3), function(k) {
  cbind(p[k, ], p[1L+k, ], p[2L+k, ], p[3L+k, ])
})

# tetrahedra meshes
tetrahedra <- lapply(1:(n-3), function(k) {
  tmesh3d(
    vertices = VS[[k]],
    indices  = th0
  )
})

# octahedra meshes
octahedra <- lapply(1:(n-3), function(k) {
  octaFromTetra(VS[[k]])
})

# non-duplicated octahedra vertices
ovs <- unique(do.call(
  rbind, 
  lapply(1:(n-3), function(k) {
    vs <- VS[[k]]
    p1 <- (vs[, 1] + vs[, 2]) / 2
    p2 <- (vs[, 1] + vs[, 3]) / 2
    p3 <- (vs[, 1] + vs[, 4]) / 2
    p4 <- (vs[, 2] + vs[, 3]) / 2
    p5 <- (vs[, 2] + vs[, 4]) / 2
    p6 <- (vs[, 3] + vs[, 4]) / 2
    rbind(p1, p2, p3, p4, p5, p6)
  })
))

# plot
open3d(windowRect = 50 + c(0, 0, 512, 512))
view3d(0, 90)
invisible(lapply(1:(n-3), function(k) {
  wire3d(tetrahedra[[k]], color = "navy", lwd = 4)
  shade3d(octahedra[[k]], color = "darkgreen", polygon_offset = 1)
  wire3d(octahedra[[k]], color = "black", lwd = 4)
  invisible()
}))
spheres3d(p, radius = 0.1, color = "navy")
spheres3d(ovs, radius = 0.06, color = "darkgreen")


# animation ####
movie3d(
  spin3d(axis = c(0, 0, 1), rpm = 60),
  duration = 1, fps = 60,
  movie = "zzpic", dir = ".",
  convert = FALSE, webshot = FALSE,
  startTime = 1/60
)

library(gifski)
gifski(
  Sys.glob("zzpic*.png"),
  "Octahelix.gif",
  width = 512, height = 512,
  delay = 1/10
)
file.remove(Sys.glob("zzpic*.png"))

Tetrahelix.R

library(rgl)

# tetrahedra vertices
r <- 3*sqrt(3)/10
theta <- acos(-2/3)
h <- 1/sqrt(10)
n <- 20
i <- 1:n
p <- cbind(r*cos(i*theta), r*sin(i*theta), i*h)

# we will also plot the circular helix
ii <- seq(1, n, length.out = 600L)
helix <- cbind(r*cos(ii*theta), r*sin(ii*theta), ii*h)
tube <- addNormals(cylinder3d(
  helix, radius = 0.05, sides = 30, closed = TRUE
))

# take the tetrahedron faces
th0 <- tetrahedron3d()$it

# there are n-3 tetrahedra in the helix
tetrahedra <- lapply(1:(n-3), function(k) {
  vs <- cbind(p[k, ], p[1+k, ], p[2+k, ], p[3+k, ])
  tmesh3d(
    vertices = vs,
    indices  = th0
  )
})

# plot
open3d(windowRect = 50 + c(0, 0, 512, 512))
colors <- viridisLite::viridis(n-3)
invisible(lapply(1:(n-3), function(k) {
  shade3d(tetrahedra[[k]], color = colors[k], polygon_offset = 1)
  wire3d(tetrahedra[[k]])
  invisible()
}))
shade3d(tube, color = "maroon")
spheres3d(p, radius = 0.1, color = "black")


# animation ####
movie3d(
  spin3d(axis = c(0, 0, 1), rpm = 60),
  duration = 1, fps = 60,
  movie = "zzpic", dir = ".",
  convert = FALSE, webshot = FALSE,
  startTime = 1/60
)

library(gifski)
gifski(
  Sys.glob("zzpic*.png"),
  "BoerdijkCoxeterTetrahelix.gif",
  width = 512, height = 512,
  delay = 1/10
)
file.remove(Sys.glob("zzpic*.png"))