Equilateral Triangle With Lattice Points

Geometry Level 5

There are 27 points, and their coordinates are ( x , y , z ) (x,y,z) , where x , y , z ∈ -1,0,1 x,y,z\in\text{{-1,0,1}}

I choose 3 points from them, and they form an equilateral triangle.

How many equilateral triangles can be formed like this?

Bonus: This is the 3 × 3 × 3 3\times3\times3 case. Generalize the n × n × n n\times n\times n case.


The answer is 80.

X X by X X , 17, Taiwan

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1 solution

Generate the lattice points.

p = Flatten [ Table [ { x , y , z } , { x , − 1 , 1 } , { y , − 1 , 1 } , { z , − 1 , 1 } ] , 2 ] ; p=\text{Flatten}[\text{Table}[\{x,y,z\},\{x,-1,1\},\{y,-1,1\},\{z,-1,1\}],2];

Generate the triangles.

t = Flatten [ Table [ { p [ [ i ] ] , p [ [ j ] ] , p [ [ k ] ] } , { i , 1 , Length [ p ] } , { j , i + 1 , Length [ p ] } , { k , j + 1 , Length [ p ] } ] , 2 ] ; t=\text{Flatten}[\text{Table}[\{p[[i]],p[[j]],p[[k]]\},\{i,1,\text{Length}[p]\},\{j,i+1,\text{Length}[p]\},\{k,j+1,\text{Length}[p]\}],2];

Write a test for a triangle being equilateral.

equilateralQ = SquaredEuclideanDistance [ $#$1 [ [ 1 ] ] , $#$1 [ [ 2 ] ] ] = SquaredEuclideanDistance [ $#$1 [ [ 1 ] ] , $#$1 [ [ 3 ] ] ] ∧ SquaredEuclideanDistance [ $#$1 [ [ 1 ] ] , $#$1 [ [ 2 ] ] ] = SquaredEuclideanDistance [ $#$1 [ [ 2 ] ] , $#$1 [ [ 3 ] ] ] & ; \text{equilateralQ}= \\ \ \ \text{SquaredEuclideanDistance}[\text{\$\#\$1}[[1]],\text{\$\#\$1}[[2]]]=\text{SquaredEuclideanDistance}[\text{\$\#\$1}[[1]],\text{\$\#\$1}[[3]]]\land \\ \ \ \ \ \text{SquaredEuclideanDistance}[\text{\$\#\$1}[[1]],\text{\$\#\$1}[[2]]]=\text{SquaredEuclideanDistance}[\text{\$\#\$1}[[2]],\text{\$\#\$1}[[3]]]\&;

Apply test triangles and count the successful cases.

Length [ Select [ t , equilateralQ ( $#$1 ) & ] ] ⟹ 80 \text{Length}[\text{Select}[t,\text{equilateralQ}(\text{\$\#\$1})\&]] \Longrightarrow 80

No overt recursion was required. This could have been done in most computer languages.

0 Helpful 0 Interesting 0 Brilliant 2 Confused

I can only find, 64 ( 2 ) \sqrt(2) triangles, and 8 2 ( 2 ) 2 \sqrt(2) sized ones, what am I missing?

Alex Burgess - 2 years, 1 month ago

1 pending report

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