Euler-Poincaré-Characteristic of Polyhedra

Question

Given a triangulation of the surface of a polyhedron p, calculate its Euler-Poincaré-Characteristic χ(p) = V-E+F, where V is the number of vertices, E the number of edges and F the number of faces.

Details

The vertices are enumerated as 1,2,...,V. The triangulation is given as a list, where each entry is a list of the vertices of one face, given in clockwise or counterclockwise order.

Despite the name, the triangulation can also contain faces with more than 3 sides. The faces can assumed to be simply connected that means that the boundary of each faces can be drawn using one closed non-self-intersecting loop.

Examples

Tetrahedron: This tetrahedron is convex and has χ = 2. A possible triangulation is

[[1,2,3], [1,3,4], [1,2,4], [2,3,4]]

Cube: This cube is convex and has χ = 2. A possible triangulation is

[[1,2,3,4], [1,4,8,5], [1,2,6,5], [2,3,7,6], [4,3,7,8],  [5,6,7,8]]

Donut: This donut/toroid shape has χ = 0. A possible triangulation is

[[1,2,5,4], [2,5,6,3], [1,3,6,4], [1,2,7,9], [2,3,8,7], [1,9,8,3], [4,9,8,6], [4,5,7,9], [5,7,8,6]]

Double Donut: This double-donut should have χ = -2. It is constructed by using two copies of the donut above and identifying the sides [1,2,5,4] of the first one with the side [1,3,6,4] of the second one.

[[2,5,6,3], [1,3,6,4], [1,2,7,9], [2,3,8,7], [1,9,8,3], [4,9,8,6], [4,5,7,9], [5,7,8,6], [1,10,11,4], [10,11,5,2], [1,10,12,14], [10,2,13,12], [1,14,13,2], [4,14,13,5], [4,11,12,14], [11,12,13,5]]

(Examples verified using this Haskell program.)

Answer 1

Haskell, 42 bytes

f m=maximum(id=<<m)-sum[0.5|_:_:l<-m,x<-l]

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Combines the face and edge terms by subtracting 0.5 for every edge on a face beyond the first two.

Alt 42 bytes:

f m=maximum(id=<<m)-sum(0.5<$(drop 2=<<m))

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Answer 2

Haskell, 49 46 bytes

u=length
f x|j<-x>>=id=maximum j+u x-u j`div`2

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I get the number of vertices by concating the faces and finding the maximum. I find the number of faces by taking the length. I find the number of edges by summing the lengths of the faces and dividing by 2.

Answer 3

APL (Dyalog Classic), 13 bytes

⌈/∘∊+≢-2÷⍨≢∘∊

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Answer 4

Jelly, 18 17 11 10 9 bytes

1 byte thanks to Erik the Outgolfer, and 1 more for telling me about Ɗ.

FṀ_FLHƊ+L

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Uses the actually intelligent not-hacked-together solution everybody else is probably using. (Credit to @totallyhuman for the only other solution I could understand enough to reimplement it.)

Old solution (17 bytes)

ṙ€1FżFṢ€QL
;FQL_Ç

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I hope I got everything right. Assumes that all faces contain at least 3 vertices and that no two faces have the same vertices; I'm not good enough in topology to come up with something that breaks the code.

Alternative 17 byte solution:

ṙ€1FżFṢ€,;F$QL$€I

Explanation

;FQL_Ç    Main link. Argument: faces
            e.g. [[1,2,3],[1,3,4],[1,2,4],[2,3,4]]
 F          Flatten the list. We now have a flat list of vertices.
            e.g. [1,2,3,1,3,4,1,2,4,2,3,4]
;           Append this to the original list.
            e.g. [[1,2,3],[1,3,4],[1,2,4],[2,3,4],1,2,3,1,3,4,1,2,4,2,3,4]
  Q         Remove duplicates. We now have a list of faces and vertices.
            e.g. [[1,2,3],[1,3,4],[1,2,4],[2,3,4],1,2,3,4]
   L        Get the length of this list. This is equal to V+F.
            e.g. 8
     Ç      Call the helper link on the faces to get E.
            e.g. 6
    _       Subtract the edges from the previous result to get V-E+F.
            e.g. 2

ṙ€1FżFṢ€QL    Helper link. Argument: faces
                e.g. [[1,2,3],[1,3,4],[1,2,4],[2,3,4]]
ṙ€1             Rotate each face 1 position to the left.
                e.g. [[2,3,1],[3,4,1],[2,4,1],[3,4,2]]
   F            Flatten this result.
                e.g. [2,3,1,3,4,1,2,4,1,3,4,2]
     F          Flatten the original faces.
                e.g. [1,2,3,1,3,4,1,2,4,2,3,4]
    ż           Pair the items of the two flattened lists.
                e.g. [[2,1],[3,2],[1,3],[3,1],[4,3],[1,4],[2,1],[4,2],[1,4],[3,2],[4,3],[2,4]]
      Ṣ€        Order each edge.
                e.g. [[1,2],[2,3],[1,3],[1,3],[3,4],[1,4],[1,2],[2,4],[1,4],[2,3],[3,4],[2,4]]
        Q       Remove duplicates. We now have a list of edges.
                e.g. [[1,2],[2,3],[1,3],[3,4],[1,4],[2,4]]
         L      Get the length of the list to get E.
                e.g. 6

Answer 5

Perl 5 -a, 29 bytes

This one is tailor made for the perl -a option which does almost all the work already

#!/usr/bin/perl -a
@V[@F]=$e+=@F}{say$#V+$.-$e/2

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Answer 6

Python 2, 47 bytes

-1 byte thanks to... user56656 (was Wheat Wizard originally).

lambda l:len(l)-len(sum(l,[]))/2+max(sum(l,[]))

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Answer 7

Wolfram Language (Mathematica), 24 bytes

-3 bytes thanks to @att.

Max@#-Tr[Tr/@(1^#)/2-1]&

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Answer 8

Stax, 9 bytes

ÑF4╨Ω◙╜#├

Run and debug online

It's a straight-forward port of totallyhuman's python solution.

%   length of input (a)
x$Y flatten input and store in y
%h  half of flattened length (b)
-   subtract a - b (c)
y|M maximum value in y (d)
+   add c + d and output

Run this one

Answer 9

Pari/GP, 28 bytes

x->#x+#Set(y=concat(x))-#y/2

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Answer 10

05AB1E, 10 9 bytes

ZsgI˜g;-+

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Explanation

Z          # push number of vertices (V)
 sg        # push number of faces (F)
   I˜g;    # push number of edges (E)
       -   # subtract (F-E)
        +  # add (F-E+V)

Answer 11

Pyth, 12 bytes

+-eSsQ/lsQ2l

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Answer 12

Scala 3, 46 bytes

A port of @Wheat Wizard♦'s Haskell answer in Scala.

---

x=>{val j=x.flatten;j.max+x.length-j.length/2}

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Answer 13

K (ngn/k), 22 bytes

{(#x)+(|/y)--2!#y}/,/\

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Answer 14

Ruby, 35 bytes

->a{a.size+a.flatten!.max-a.size/2}

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Answer 15

Proton, 35 bytes

l=>max(f=sum(l,[]))-len(f)/2+len(l)

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Answer 16

JavaScript (ES6), 60 bytes

a=>a.map(b=>(v=Math.max(v,...b),d+=b.length/2-1),d=v=0)&&v-d

Explanation: Loops over each face, keeping track of the largest vertex seen in v and tracking the number of edges minus the number of faces in d as per @xnor's answer.