Vertices of a regular dodecahedron

Question

A regular dodecahedron is one of the five Platonic solids. It has 12 pentagonal faces, 20 vertices, and 30 edges.

Your task is to output the vertex coordinates of a regular dodecahedron. The size, orientation, and position of the dodecahedron are up to you, as long as it is regular.

You may output the coordinates in any order, and in any reasonable format.

If the edge length of your dodecahedron is \$a\$, then the coordinates should be accurate to at least \$a/1000\$.

This is code-golf, so the shortest code in bytes wins.

Example output

This is one possible output. Your output does not have to match this.

{-1.37638,0.,0.262866}
{1.37638,0.,-0.262866}
{-0.425325,-1.30902,0.262866}
{-0.425325,1.30902,0.262866}
{1.11352,-0.809017,0.262866}
{1.11352,0.809017,0.262866}
{-0.262866,-0.809017,1.11352}
{-0.262866,0.809017,1.11352}
{-0.688191,-0.5,-1.11352}
{-0.688191,0.5,-1.11352}
{0.688191,-0.5,1.11352}
{0.688191,0.5,1.11352}
{0.850651,0.,-1.11352}
{-1.11352,-0.809017,-0.262866}
{-1.11352,0.809017,-0.262866}
{-0.850651,0.,1.11352}
{0.262866,-0.809017,-1.11352}
{0.262866,0.809017,-1.11352}
{0.425325,-1.30902,-0.262866}
{0.425325,1.30902,-0.262866}

Answer 1

Wolfram Language (Mathematica), 15 bytes

SpherePoints@20

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20 evenly spaced points on the surface of a unit sphere.

The more obvious PolyhedronCoordinates@Dodecahedron[] is a little bit longer.

Answer 2

05AB1E, 31 25 bytes

®X‚D3ãsãvy5t<;D>‚*¾š2Ý._«

Uses \$\sqrt{3}\$ as the distance of each vector from the origin \$\{0,0\}\$. Will output the following 20 vertex-coordinates:

$$[\{-1,-1,-1\},\{-1,-1,1\},\{-1,1,-1\},\{-1,1,1\},\{1,-1,-1\},\{1,-1,1\},\{1,1,-1\},\{1,1,1\},\{0,-\phi,-\phi-1\},\{-\phi,-\phi-1,0\},\{-\phi-1,0,-\phi\},\{0,-\phi,\phi+1\},\{-\phi,\phi+1,0\},\{\phi+1,0,-\phi\},\{0,\phi,-\phi-1\},\{\phi,-\phi-1,0\},\{-\phi-1,0,\phi\},\{0,\phi,\phi+1\},\{\phi,\phi+1,0\},\{\phi+1,0,\phi\}]$$

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Explanation:

®X‚             # Push pair [-1,1]
   D            # Duplicate it
    3ã          # Get all possible triplets using the cartesian power of 3
   s            # Swap so the [-1,1] pair is at the top again
    ã           # Also get all possible pairs using the cartesian power of 2
     vy         # Loop over each pair of [±1,±1]:
       5t<;     #  Push the golden ratio: (sqrt(5)-1)/2
           D    #  Duplicate it
            >   #  Increase the copy by 1
             ‚  #  Pair it together with the (sqrt(5)-1)/2
      y       * #  Multiply the values in the pair to pair [±1,±1] at the same positions
        ¾š      #  Prepend a 0 to this pair
          2Ý    #  Push list [0,1,2]
            ._  #  Rotate the triplet that many times to the left
              « #  Merge it to the original list of [-1,1]-triplets
                # (after which the list of 20 triplets is output implicitly as result)

---

I started with a port of this SO C# answer, but noticed (for \$r=\sqrt{3}\$ at least):

• \$±\frac{1}{\sqrt{3}}\times\phi\times\sqrt{3}\$ is simply \$±\phi\$
• \$±\frac{\frac{1}{\sqrt{3}}}{\phi}\times\sqrt{3}\$ is simply \$±(\phi+1)\$

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Answer 3

Python, 73 bytes

x={3*(55,),(89,34,0)}
for s in(-1,1)*3:x|={(s*y,*z)for*z,y in x}
print(x)

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Same approach as @xnor (I think), i.e. flip signs and rotate coordinates random-ish-ly. Only, I'm being more simple-minded about it ;-)

Retired Python, 103 bytes

*x,x[7],x[2]=6*[-55,55]
*y,y[11],_,_,y[2]=3*[-34,0,-89,34,0,89]
for a in x,y:*map(print,a,a[1:],a[2:]),

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Integer coordinates from @alephalpha / @xnor via @Arnauld.

Answer 4

JavaScript (ES6), 88 bytes

Returns a set of strings, where each string is a triplet x,y,z.

This version takes inspiration from xnor's approach.

_=>(g=(s,x,y=x,z=x,k=y)=>k--?g(s.add(x+[,y,z]),k%4?z:-z,x,y,k):s)(g(new Set,55),0,34,89)

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Commented

_ => (               // main function ignoring its argument
  g = (              // g is a helper function taking:
    s,               //   s = output set
    x,               //   x = 1st coordinate
    y = x,           //   y = 2nd coordinate, or x by default
    z = x,           //   z = 3rd coordinate, or x by default
    k = y            //   k = counter, initialized to y
  ) =>               //
  k-- ?              // if k is not 0 (decrement afterwards):
    g(               //   do a recursive call:
      s.add(         //     add to the set:
        x + [, y, z] //       a stringified version of the triplet
      ),             //
      k % 4 ? z      //     rotate by putting z at the beginning
            : - z,   //     or -z if k is a multiple of 4
      x, y,          //     followed by x and y
      k              //     pass the updated counter
    )                //   end of recursive call
  :                  // else:
    s                //   stop and return the set
)(                   //
  g(new Set, 55),    // first call with x = y = z = 55
  0, 34, 89          // second call with (x, y, z) = (0, 34, 89)
)                    //

---

JavaScript (V8), 108 bytes

-5 bytes by using \$\phi\approx 89/55\$, as suggested by alephalpha
-3 bytes by scaling everything up (x55), as suggested by xnor

A full program printing 20 triplets x,y,z.

[344,900,564,228].map(g=n=>n--&15&&g(n,j=0,print([4,6,8].map(i=>[0,55,89,34][v=n>>i&3]*(v&&n>>j++&1||-1)))))

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Encoding

We use the Cartesian coordinates based on the golden ration as described on Wikipedia, but with a different scale.

Each triplet pattern is encoded with the following bit mask:

ZZ YY XX NNNN
 |  |  | \__/
 |  |  |   |
 |  |  |   +--> number of triplets
 |  |  +------> x
 |  +---------> y
 +------------> z

where \$x\$, \$y\$ and \$z\$ are indices in the lookup array \$[0,1,\phi,\frac{1}{\phi}]\$.

This gives:

Triplet pattern	x	y	z	N	Bit mask	As decimal
\$(\pm 1,\pm 1,\pm 1)\$	1	1	1	8	01 01 01 1000	344
\$(0,\pm\phi,\pm\frac{1}{\phi})\$	0	2	3	4	11 10 00 0100	900
\$(\pm\frac{1}{\phi},0,\pm\phi)\$	3	0	2	4	10 00 11 0100	564
\$(\pm\phi,\pm\frac{1}{\phi},0)\$	2	3	0	4	00 11 10 0100	228

Answer 5

Haskell, 33 bytes

k=[0..19]

[[0^(x-y)^2|x<-k]|y<-k]

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Explanation

So this answer uses a trick, which some may consider a loophole. Actually uses two tricks. So I'm going to explain how it works.

If the OP wants to patch this approach out of the challenge, I will delete my answer.

Skew polyhedra

The exact definition of a polyhedron varies from person to person. Some circumstances require that a polyhedron be finite and not self intersect, this gives 5 regular polyhedra, some allow self intersection but still require finiteness, this gives 9 regular polyhedra, some times Euclidean tilings are considered as well giving 12.

However in the beginning of the 20th century Petrie and Coxeter discovered 3 regular polyhedra that hadn't been considered before. These were infinite polyhedra with flat non-intersecting faces, however they didn't have insides, so they hadn't been considered before. Pictured below is a section of the mucube, one of the three:

This discovery lead to a new definition of polytope (and by extension a new definition of polyhedron). This definition was first given by Branko Grünbaum in the paper Regular polyhedra - old and new, but has been used pretty extensively. Under this definition vertices (and edges) have exact locations, but the higher dimensional elements no longer needed to have exact interiors. A polygon is just a sequence of vertices pairwise connected by edges, such that you can reach any vertex from any other by following edges. A polyhedron is shape made out of polygons with exactly two meeting at every edge.

For example, classically a polygon needs to be "flat" meaning all of its vertices lie in a plane. However skew polygons can have vertices which lie in higher dimensional space. For example here's a 4-dimensional regular polygon:

Image by Plasmath see this page for liscensing

(We will come back to this particular polygon later)

The same can also be done with polyhedra. Instead of requiring that the vertices of a polyhedron lie strictly in 3D space with each face lying on a plane, you can have polyhedra whose vertices span 4 or greater dimensions, and whose faces are non-planar.

Lots of great work would be done with this definition. I'd especially like to point out the work of McMullen and Schulte. The have a paper classifying all the regular polyhedra in 3-space, which has been summarized in this excellent youtube video:

jan Misali, there are 48 regular polyhedra

This definition is the one I am using in this answer.

Now the challenge says:

Your task is to output the vertex coordinates of a regular dodecahedron. The size, orientation, and position of the dodecahedron are up to you, as long as it is regular.

So I just need to choose a regular dodecahedron. It turns out there are 28 distinct ways to realize the regular dodecahedron {5,3} in Euclidean space. The dodecahedron I chose is the its "Simplex realization". In general you can take any regular polyhedron with \$n\$ vertices and create an fully symmetric realization of it on a \$(n-1)\$-simplex. This is because all the symmetries of a polyhedron are permutations of its vertices. The symmetries of a simplex are all the permutations of its vertices. So when we place the vertices of a polyhedron on the vertices of a simplex every symmetry is possible. So this realization is always fully regular with maximum symmetry.

The dodecahedron has 20 vertices so that means the vertices of its simplex realization are the vertices of the 19-simplex. At first glance I've just made my problem much harder. The vertices of the 19-simplex centered at the origin are nasty, way worse than the vertices of the convex regular dodecahedron.

So here's where the second trick comes in. The vertices of a regular \$n\$-simplex are actually really simple if you express them in \$n+1\$ dimensions. If you place each vertex on an axis, all at the same distance from the origin, then permuting the axes is exactly the same as permuting the vertices. So this is fully regular, in fact its the same simplex its just lying on a weird hyperplane.

For example you can give the vertices of the triangle as:

$$ (1,0,0) \\ (0,1,0) \\ (0,0,1) \\ $$

This is a regular 2D triangle. This same trick comes up if you want to represent triangular (or hexagonal) coordinates. Instead of using 2D space and complicated square roots, you can represent them in 3D space as triplets of integers.

So returning to the problem at hand, the 19 simplex can be given in 20 dimensions as all permutations of \$(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)\$. Its symmetries being all permutations of of the coordinate axes.

So in summary, I am giving the vertices for a 19-dimensional dodecahedron in 20 dimensional space.

I can't exactly render this in any way that is helpful. But its faces I can say its faces are the regular 4D pentagon shown earlier. If you want to try to imagine 12 of those in 20 dimensional space.

Answer 6

Python 3, 86 bytes

r=[]
for l in[[55]*3,[0,89,34]]*99:l+=(len(r)%11%-2|1)*l.pop(0),;r+=(*l,),
print({*r})

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Outputs the golden-ratio vertices everyone is using, scaled up by 55, from the approximation of \$ \phi \approx 89/55\$ suggested by alephalpha.

The idea is to take the two vertices \$(1,1,1)\$ and \$(0,\phi,1/\phi)\$ and "randomly" negate entries and take cyclic permutations to reach all 20 vertices, then de-duplicate using sets. Each loop alternates between the two vertices, mutating it by moving the first entry to the back, and possibly negating that entry. Whether to negate is given by i%11%2 on the i'th loop, which happens to hit all 20 vertices in not too many loops. The index i is actually tracked by the length of the list l of vertices we've generated so far.

---

Python 3, 82 bytes

*l,k=0,34,89,718230
while k:k-175or(l:=[55]*3);l+=l.pop(0)*(-1)**k,;k>>=1;print(l)

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To be golfed...

Answer 7

Prolog (SWI), 130 bytes

A+B:-A=B;A is-B.
A*B*C*G*I*P:-A+P,B+P,C+P;A=0,B+G,C+I;B=0,A+I,C+G;C=0,A+G,B+I.
:-bagof([A,B,C],A*B*C*89*34*55,R),maplist(write,R).

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Uses the approximation \$\phi \approx 89/55\$ then scaling the vertices by 55.

Answer 8

Nekomata, 16 bytes

7Ƃ89\55:ŗÐçxŘ~?ŋ

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7Ƃ89\55:ŗÐçxŘ~?ŋ
7Ƃ                  Push the list [1,1,1]
  89\55             Push the number 89/55
       :ŗÐ          Pair with its reciprocal ([89/55,55/89])
          ç         Prepend zero ([0,89/55,55/89])
           xŘ~      Non-deterministic rotation ([0,89/55,55/89],[89/55,55/89,0],[55/89,0,89/55])
              ?     Non-deterministic choice ([1,1,1],[0,89/55,55/89],[89/55,55/89,0],[55/89,0,89/55])
               ŋ    Optionally negate each element

Answer 9

Perl, 167 bytes

sub f{print"@_
"}sub g{$a=shift;map{$_[$_]*(!($a&1<<$_)*2-1)}0..~~@_-1}$p=.5+sqrt 5/4;f g $_,1,1,1 for 0..7;do{@v=((g $_,$p,1/$p,0)x2);f @v[$_..$_+2] for 0..2}for 0..3

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Answer 10

Jelly,  18  17 bytes

^{...or 15 with repeats - remove µQ.}

Ø+ṗ3µØpİƬŻṙJ×€jµQ

A niladic Link that yields a list of triples.

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How?

Produces the twenty triples as shown on Wikipedia:

$$(\pm 1, \pm 1, \pm 1)$$ $$(0, \pm \phi, \pm \phi^{-1})$$ $$(\pm \phi^{-1}, 0, \pm \phi)$$ $$(\pm \phi, \pm \phi^{-1}, 0)$$

Ø+ṗ3µØpİƬŻṙJ×€jµQ - Link: no arguments
Ø+                - [1, -1]
   3              - three
  ṗ               - Cartesian power -> (+/-1, +/-1, +/-1) "CubePoints"
    µ          µ  - monadic chain - f(CubePoints):
     Øp           -   phi
        Ƭ         -   collect up while distinct under:
       İ          -     inverse
         Ż        -   prefix with zero -> [0, phi, 1/phi]
           J      -   range of length {CubePoints} -> [1, 2, 3, 4, 5, 6, 7, 8]
          ṙ       -   rotate {[0, phi, 1/phi]} left by (vectorises) {that}
                      -> [[phi, 1/phi, 0], [1/phi, 0, phi], [0, phi, 1/phi], ...(8)]
             €    -   for each (rotation):
            ×     -     multiply by {CubePoints} (vectorises)
              j   -   join {that list of lists of triples} with {CubePoints}
                Q - deduplicate

Answer 11

Charcoal, 53 44 bytes

≔Ｅ²Ｅ³∨ι∧⊖λ⊘⁺⊖λ₂⁵υＦυＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰Ｆ¬№υκ⊞υκＩυ

Try it online! Link is to verbose version of code. Explanation: Uses the formula from this Math.SE answer.

≔Ｅ²Ｅ³∨ι∧⊖λ⊘⁺⊖λ₂⁵υ

Get the points [1/ϕ, 0, ϕ] and [1, 1, 1].

ＦυＦＥ²⊞ＯΦιν×⊖⊗κ§ι⁰Ｆ¬№υκ⊞υκＩυ

For each point, rotate it around the axis x=y=z, and also reflect the rotated point in the xy plane, and save and process any new points thus found.

Ｉυ

Output the final dodecahedron.

Answer 12

Ruby, 55 bytes

20.times{|j|p ~0**j*(1.309-k=j/10),1.618**k*1i**j*=0.4}

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Golfed version using decimal approximations instead of exact formulas (I can't use fractions as Ruby would interpret using integer arithmetic.) Dimensions are now half the ones given in the wolfram page, instead of double, to avoid j/10*4. The polar vertices are now listed before the equatorial vertices, to enable the distance from the z axis to be calculated as 1.618**k (1 or 1.618)

I also deleted the [] to save 2 bytes, although this impacts readability of the output.

Ruby, 85 63 bytes

20.times{|j|p [~0**j*z=5**0.5-1+j/10*4,[8/z,4].max*1i**j*=0.4]}

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The coordinates given in the question are x,y,z with the dodecahedron's 5-fold rotational symmetry axis aligned with the z axis (It's actually a 10-fold axis for the rotate-and-flip operation.). This program outputs the same coordinates as the ones in the question, but in the format [z,x+yi] and scaled. First all the equatorial vertices are output, and then the polar vertices.

~0**j alternates the sign of the z coordinate as the x and y coordinates cycle through the 1/10th of a circle rotations. (-1**j would not work because unary minus has lower priority than raising to a power in Ruby.)

Exact coordinates in terms of the golden ratio are given at the wolfram page in formulas 1 and 2. The alternate coordinate system used in other answers is given a little below them on the same page, as well as in wikipedia. This code uses double the scale used in the Wolfram page, to avoid the division by 2 required to calculate the golden ratio. The footer in TIO link scales the coordinates given in the question and moves the z coordinate to the beginning for comparison. You still have to squint a bit because I've sorted the coordinates given in the question by z value, whereas the above code generates the vertices in an intercalated order with alternating sign of the z value.

Answer 13

Pyth, 24 bytes

{+K^_BT3*VM*K.<L+tB.n3Z3

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Uses the same \$\varphi\$ based coordinates that everybody else is using, just multiplied by 10, that is, \$[\pm 10,\pm 10,\pm 10], [\pm 10\varphi, \pm \frac{10}{\varphi}, 0], [0, \pm 10\varphi, \pm \frac{10}{\varphi}], [\pm \frac{10}{\varphi}, 0, \pm 10\varphi]\$.

But there's a really neat simplification here, which is that after I had the integer coordinates, I could use them as the signs for the other coordinates, just adding the need to deduplicate at the end since \$0=-0\$.

Explanation

    _BT                     # bifurcate 10 on negation (gives us [1,-1])
   ^   3                    # get all possible triplets
  K                         # assign this list to K
                   .n3      # phi
                 tB         # bifurcated on subtracting 1
                +     Z     # append 0 to the pair
             .<L       3    # map [0,1,2] to cyclic rotation on this list
           *K               # get the cartesian product of these rotations with K
        *VM                 # map over vectorized multiplication
 +K                         # add K to this list
{                           # deduplicate to eliminate repeat coordinates

Answer 14

Scala, 226 bytes

Golfed version. Try it online!

val p=Seq(-1,1)
val g=(sqrt(5)-1)/2     
((for{a<-p;b<-p;c<-p}yield Seq(a,b,c))++(for{p<-((for{a<-p;b<-p}yield Seq(a,b)).map{case Seq(a,b)=>Seq(a*g,b*(g+1))}.map(0.0+:_));i<-0 to 2}yield p.drop(i)++p.take(i))).foreach(println)

Ungolfed version. Try it online!

import scala.math._

object Main {
  def main(args: Array[String]): Unit = {
    val pair = List(-1, 1)
    val triplets = for {
      a <- pair
      b <- pair
      c <- pair
    } yield List(a, b, c)
    val pairs = for {
      a <- pair
      b <- pair
    } yield List(a, b)
    val goldenRatio = (sqrt(5) - 1) / 2
    val goldenRatioMultipliedPairs = pairs.map {
      case List(a, b) => List(a * goldenRatio, b * (goldenRatio + 1))
    }
    val prependedPairs = goldenRatioMultipliedPairs.map(0.0 :: _)
    val rotatedPairs = for {
      pair <- prependedPairs
      i <- 0 to 2
    } yield pair.drop(i) ++ pair.take(i)
    val finalTriplets = triplets ++ rotatedPairs
    finalTriplets.foreach(println)
  }
}

Answer 15

Perl, 139 137 135 bytes

sub p{print"@_
"}sub g{($x)=@_;map{$_[$_]*(1-(4*$x>>$_&2))}1..3}p g$_,(55)x3 for 0..7;map{@v=(g$_,89,34,0)x2;p@v[$_..$_+2]for 0..2}0..3

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