Goossen Karssenberg recently asked if this dissection of an equilateral triangle into five similar triangles was known:
I know a few geometers who knew of this, but I’m not sure any of us have published anything about it. The base triangles is in the algebraic field of the supergolden ratio, ψ. I discuss the triangle at my blog post Shattering the Plane, noting that it has an angle of 120°.
The supergolden ratio solves this equation:
In[]:=
Solve[ψ^3==ψ^2+1]
Out[]=
ψ,ψ,ψ
In[]:=
ψ=;
Whenever I’m dealing with geometric constraints, I look up the discriminant of the polynomials involve to find the algebraic space.
In[]:=
NumberFieldDiscriminant[ψ]
Out[]=
-31
Here are coordinates for the dissection:
In[]:=
points=,,0,,,,,,,,,;
In the algebraic space of ψ, the coordinates are friendlier.
In[]:=
ResourceFunction["SqrtSpace"][ψ,points]
Out[]=
,,,-,-,-,{0,0,0},,,,-,-,-,-,-,-,1,-1,,,-,-,0,-,,-,,-,-,,0,-,,
1
4
1
4
1
4
1
12
1
12
1
12
1
3
1
3
1
3
1
4
1
4
1
4
1
12
1
12
1
12
1
4
2
3
1
3
1
12
3
4
1
2
1
3
17
12
5
6
1
4
1
4
5
12
1
12
1
3
Here are the edges that matter. The last value is the distance between the points as a power of ψ.
In[]:=
edges=Select#,ResourceFunction["IntegerChop"]Log,N[EuclideanDistance@@points[[#]]]&/@Subsets[Range[6],{2}],IntegerQ[Last[#]]&
Out[]=
{{{1,2},2},{{1,3},2},{{1,4},0},{{1,6},1},{{2,3},2},{{2,5},1},{{2,6},-1},{{3,4},1},{{3,5},0},{{4,5},-2},{{4,6},-1},{{5,6},0}}
And a diagram of the dissection, with edges indicating a power of ψ.
In[]:=
Graphics[{Point[points],Table[{Line[points[[edges[[k,1]]]]],Style[Text[edges[[k,2]],Mean[points[[edges[[k,1]]]]]],16]},{k,1,Length[edges]}]}]
Out[]=
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
The Psi 0-1-2 triangle dissection
by Ed Pegg
Wolfram Community, STAFF PICKS, March 1, 2024
https://community.wolfram.com/groups/-/m/t/3133636
by Ed Pegg
Wolfram Community, STAFF PICKS, March 1, 2024
https://community.wolfram.com/groups/-/m/t/3133636