Torus knots 4D

TorusKnot4D.R

library(rgl)
library(cgalMeshes)

# trefoil (2,3) ####
p <- function(theta, phi) {
  c(
    cos(theta)*cos(phi), cos(theta)*sin(phi), 
    sin(theta)*cos(1.5*phi), sin(theta)*sin(1.5*phi)
  )
}

s <- Vectorize(function(theta, phi) {
  q <- p(theta, phi)
  q[c(1L,2L,3L)] / (1 - q[4L])
})


phi_ <- seq(0, 4*pi, length.out = 400L)
pts <- t(s(0, phi_))
tube1 <- addNormals(
  cylinder3d(pts, radius = 0.05, sides = 30),
)
pts <- t(s(pi/12, phi_))
tube2 <- addNormals(
  cylinder3d(pts, radius = 0.05, sides = 30),
)
pts <- t(s(pi/6, phi_))
tube3 <- addNormals(
  cylinder3d(pts, radius = 0.05, sides = 30),
)
pts <- t(s(pi/4, phi_))
tube4 <- addNormals(
  cylinder3d(pts, radius = 0.05, sides = 30),
)

mesh1 <- addNormals(parametricMesh(
  s, c(0, pi/12), c(0, 4*pi), periodic = c(FALSE, TRUE),
  nu = 50L, nv = 600L
))
mesh2 <- addNormals(parametricMesh(
  s, c(pi/12, pi/6), c(0, 4*pi), periodic = c(FALSE, TRUE),
  nu = 50L, nv = 600L
))
mesh3 <- addNormals(parametricMesh(
  s, c(pi/6, pi/4), c(0, 4*pi), periodic = c(FALSE, TRUE),
  nu = 50L, nv = 600L
))

open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.9)
shade3d(mesh1, color = "navy")
shade3d(mesh2, color = "yellow")
shade3d(mesh3, color = "navy")
#shade3d(tube1, color = "firebrick4")
shade3d(tube2, color = "pink")
shade3d(tube3, color = "maroon")
shade3d(tube4, color = "firebrick4")

#
theta_ <- seq(0, pi/2, length.out = 10L)[1L:6L]
phi_ <- seq(0, 4*pi, length.out = 400L)

open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.95)
for(i in seq_along(theta_)) {
  theta <- theta_[i]
  pts <- t(s(theta, phi_))
  tube <- addNormals(
    cylinder3d(pts, radius = 0.05, sides = 30),
  )
  shade3d(tube, color = "navy")
}


# (3, 11)-knot
p <- function(theta, phi) {
  c(
    cos(theta)*cos(phi), cos(theta)*sin(phi), 
    sin(theta)*cos(11/3*phi), sin(theta)*sin(11/3*phi)
  )
}

s <- Vectorize(function(theta, phi) {
  q <- p(theta, phi)
  q[c(1L,2L,3L)] / (1 - q[4L])
})

theta_ <- seq(0, pi/2, length.out = 10L)[2L:5L]
phi_ <- seq(0, 6*pi, length.out = 600L)

open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.85)
for(i in seq_along(theta_)) {
  theta <- theta_[i]
  pts <- t(s(theta, phi_))
  tube <- addNormals(
    cylinder3d(pts, radius = 0.03, sides = 30),
  )
  shade3d(tube, color = "navy")
}


# (4, 3)-knot
p <- function(theta, phi) {
  c(
    cos(theta)*cos(phi), cos(theta)*sin(phi), 
    sin(theta)*cos(3/4*phi), sin(theta)*sin(3/4*phi)
  )
}

s <- Vectorize(function(theta, phi) {
  q <- p(theta, phi)
  q[c(1L,2L,3L)] / (1 - q[4L])
})

theta_ <- seq(0, pi/2, length.out = 10L)[2L:5L]
phi_ <- seq(0, 8*pi, length.out = 600L)

open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.85)
for(i in seq_along(theta_)) {
  theta <- theta_[i]
  pts <- t(s(theta, phi_))
  tube <- addNormals(
    cylinder3d(pts, radius = 0.03, sides = 30),
  )
  shade3d(tube, color = "navy")
}








# # -- if you want an animation 
M <- par3d("userMatrix")
movie3d(
  par3dinterp(
    time = seq(0, 1, len = 9),
    userMatrix = list(
      M,
      rotate3d(M, pi, 1, 0, 0),
      rotate3d(M, pi, 1, 1, 0),
      rotate3d(M, pi, 1, 1, 1),
      rotate3d(M, pi, 0, 1, 1),
      rotate3d(M, pi, 0, 1, 0),
      rotate3d(M, pi, 1, 0, 1),
      rotate3d(M, pi, 0, 0, 1),
      M
    )
  ),
  fps = 100,
  duration = 1,
  dir = ".",
  movie = "zzpic",
  convert = FALSE, webshot = FALSE
)

library(gifski)
gifski(
  png_files = Sys.glob("zzpic*.png"),
  gif_file = "TrefoilKnots4D.gif",
  width = 512,
  height = 512,
  delay = 1/8
)
file.remove(Sys.glob("zzpic*.png"))