rhombic triacontahedron

NEWS.md

@@ -1,6 +1,6 @@
 # gyro 0.1.0.9000 (2022-02-27)
 
-New examples: gircope, truncated great dodecahedron, castellated rhombicosidodecahedral prism.
+New examples: gircope, truncated great dodecahedron, castellated rhombicosidodecahedral prism, rhombic triacontahedron.
 
 
 # gyro 0.1.0 (2022-02-03)

README.md

@@ -31,3 +31,5 @@
 ![](https://raw.githubusercontent.com/stla/gyro/main/inst/images/truncatedGreatDodecahedron.gif)
 
 ![](https://raw.githubusercontent.com/stla/gyro/main/inst/images/CastellatedRhombicosidodecahedralPrism.gif)
+
+![](https://raw.githubusercontent.com/stla/gyro/main/inst/images/rhombicTriacontahedron.gif)

inst/gyrodemos/gircope.R

@@ -56,7 +56,7 @@ s <- 0.5
 open3d(windowRect = c(50, 50, 562, 562), zoom = 0.8)
 bg3d(rgb(54, 57, 64, maxColorValue = 255))
 for(i in 1L:nrow(squares)){
- square <- squares[i,]
+ square <- squares[i, ]
  mesh1 <- gyrotriangle(
  pvertices[square[1L], ], pvertices[square[2L], ], pvertices[square[3L], ],
  s = s

inst/gyrodemos/rhombicTriacontahedron.R

@@ -0,0 +1,115 @@
+library(gyro)
+library(rgl)
+library(Rvcg)
+library(Morpho)
+library(cxhull)
+
+##~~ rhombic triacontahedron ~~##
+
+# The rhombic triacontahedron can be obtained as the convex hull
+# of a 3D projection of the 6-cube.
+
+# vertices of the 6-cube
+SixCubeVertices <- as.matrix(expand.grid(rep(list(c(-1, 1)), 6L)))
+
+# the three 3D vectors spanning the space of the projection
+phi <- (1 + sqrt(5)) / 2
+u1 <- c( 1, phi, 0, -1, phi, 0)
+u2 <- c(phi, 0, 1, phi, 0, -1)
+u3 <- c( 0, 1, phi, 0, -1, phi)
+# let's normalize them
+u1 <- u1 / sqrt(c(crossprod(u1)))
+u2 <- u2 / sqrt(c(crossprod(u2)))
+u3 <- u3 / sqrt(c(crossprod(u3)))
+
+# the projection
+proj <- function(v){
+ c(c(crossprod(v, u1)), c(crossprod(v, u2)), c(crossprod(v, u3)))
+}
+vertices <- t(apply(SixCubeVertices, 1L, proj))
+
+# now we compute the convex hull
+hull <- cxhull(vertices, triangulate = FALSE)
+edges <- hull[["edges"]]
+
+# we find 32 vertices and 30 faces:
+str(hull, max = 1L)
+# List of 5
+# $ vertices:List of 32
+# $ edges : int [1:60, 1:2] 1 1 1 9 9 9 9 10 10 13 ...
+# $ ridges :List of 60
+# $ facets :List of 30
+# $ volume : num 34.8
+hullVertices <- t(vapply(hull[["vertices"]], function(vertex){
+ vertex[["point"]]
+}, numeric(3L)))
+
+# all the faces have four vertices (each face is a rhombus):
+nvertices_per_face <- vapply(hull[["facets"]], function(face){
+ length(face[["vertices"]])
+}, integer(1L))
+all(nvertices_per_face == 4L)
+
+# to split them into two triangles, we need to order their vertices
+polygonize <- function(edges){ # function which orders the vertices
+ nedges <- nrow(edges)
+ es <- edges[1L, ]
+ i <- es[2L]
+ edges <- edges[-1L, ]
+ for(. in 1L:(nedges-2L)){
+ j <- which(apply(edges, 1L, function(e) i %in% e))
+ i <- edges[j, ][which(edges[j, ] != i)]
+ es <- c(es, i)
+ edges <- edges[-j, ]
+ }
+ es
+}
+
+rhombuses <- t(vapply(
+ hull[["facets"]],
+ function(face) polygonize(face[["edges"]]),
+ integer(4L)
+))
+
+# observe the distances between all pairs of vertices of a rhombus:
+rhombusVertices <- vertices[rhombuses[1L, ], ]
+dist(rhombusVertices)
+#> 1 2 3
+#> 2 1.414214
+#> 3 2.406004 1.414214
+#> 4 1.414214 1.486992 1.414214
+# the edges have the same length
+# (a rhombus is also called "equilateral quadrilateral")
+# but there's a long diagonal and a short diagonal
+# we will split the rhombuses along the long diagonal,
+# for a better rendering of the hyperbolic polyhedron
+
+
+# draw ####
+s <- 0.8
+open3d(windowRect = c(50, 50, 562, 562), zoom = 0.7)
+bg3d(rgb(54, 57, 64, maxColorValue = 255))
+for(i in 1L:nrow(rhombuses)){
+ rhombus <- rhombuses[i, ]
+ rhombusVertices <- vertices[rhombus, ]
+ # here is how we choose to split along the long diagonal:
+ d <- c(round(dist(rhombusVertices)))
+ if(d[5L] == 2){
+ rhombusVertices <- vertices[rev(rhombus), ]
+ }
+ mesh1 <- gyrotriangle(
+ rhombusVertices[1L, ], rhombusVertices[2L, ], rhombusVertices[3L, ], s = s
+ )
+ mesh2 <- gyrotriangle(
+ rhombusVertices[1L, ], rhombusVertices[3L, ], rhombusVertices[4L, ], s = s
+ )
+ mesh <- vcgClean(mergeMeshes(mesh1, mesh2), sel = c(0, 7), silent = TRUE)
+ shade3d(mesh, color = "darkmagenta")
+}
+for(i in 1L:nrow(edges)){
+ ids <- edges[i, ]
+ A <- vertices[ids[1L], ]; B <- vertices[ids[2L], ]
+ tube <- gyrotube(A, B, s = s, radius = 0.025)
+ shade3d(tube, color = "whitesmoke")
+}
+spheres3d(hullVertices, radius = 0.03, color = "whitesmoke")