Are there any known plastic constant triangles?

Question

I am trying to determine if there are any known plastic constant triangles. By this I mean specifically triangles for which all sides are powers of the plastic constant, $p\approx1.324717957244746$. The problem is that if you try to Google plastic constant triangles you'll find everything you wanted to know about recycling plastic, but nothing about the question I've posed here.

Answer 1

All of them are of the form $$ p^k \cdot( 1,p^a,p^{a+b}) $$ for non-negative integers $a,b,k$ with some restrictions:

(I) if $b=0$ we can allow any $a$

(II) if $b=1$ we demand $a \leq 3. \;$ NOTE $p^5-p^4 = 1.$

(III) if $b=2$ we demand $a=0. \;$ NOTE $p^3 - p = 1.$

Any $p^k$ times one of these: $$ (1,p^a,p^a); \; \; (1,1,p); \; \; (1,p,p^2); \; \; (1,p^2,p^3); \; \; (1,p^3,p^4); \; \; (1,1,p^2). $$

we cannot allow $b > 2$

Answer 2

The triangle inequality says that for a triangle of sides of lengths $a, b, c$, the following inequalities all hold: $a < b + c$ and $b < c + a$ and $c < a + b$. If $c$ is the longest side of the triangle, then we need only check that last one.

I'm assuming that you don't want an equilateral triangle with sides $p, p, p$.

So one possibility is to have an isosceles triangle with sides $1, 1, p$. This triangle is valid, because $p < 1 + 1 = 2$. The triangle with sides $p, p, 1$ also works.

A more interesting plastic triangle might have sides $1, p, p^2$. This is also a valid triangle, since $0<p<\sqrt{2}$, so $p^2 < 2 < 1 + p$

Answer 3

The golden ratio $r = \phi$ satisfies $\,1+r=r^2\,$ which implies that a triangle with sides $\,1,r,r^2\,$ is degenerate. But, for any real number $r$ such that $1/\phi<r<\phi\,$, then since $1+r > r^2, 1+r^2>r, \,$ and $r+r^2>1,\,$ implies that the triangle with sides $\,1,r,r^2\,$ is not degenerate. Applying this to $\,r=p\,$, since $\, 1/\phi<1<p<\phi\,$ the triangle with sides $\,1,p,p^2\,$ is not degenerate. Of course, you can multiply all three sides of the triangle by the same power of $p$ resulting in a triangle with sides $\,p^k,p^{k+1},p^{k+2}.\,$ There are other possibilites along the same lines and they have to satisfy the triangle inequalities.

Answer 4

You can put 19 points in 3D space so that the distance between any two is an integer power of the square root of the plastic constant. This structure contains most of the plastic-power triangles. Many of these triangles have unusual properties, some of which I cover at Shattering the Plane. Below, you can see a distance matrix for the $K_{19}$ structure.

Also, you can arrange a progressive series for a perfect rectangle dissection.