An Approach to Geometric Modeling Using Genetic Programming

Abstract

In this work, we ‘derived’ the famous Pythagorean theorem from the measurements of the sides of right-angled triangles with machine learning. In classical Euclidean geometry, this result is proved with rigorous geometrical argument, but we have followed a data-driven approach and got the same result without entering a single step in the domain of geometry. We used symbolic regression with genetic programming to reach the model. As far as our knowledge goes, this result is a novel one and may open up a new avenue of applying machine learning tool in geometry. We have used Python programming language 3.7 and libraries such as DEAP (v1.2) and pygraphviz. The whole project can be found on https://github.com/snigdhasjg/Pythagorean-Triplate.git.

Appendix

The algorithm gives output as string representation of tree like $power\left( {add\left( {mul\left( {A,A} \right),mul\left( {B,B} \right)} \right),1,2} \right)$ that can be expressed as $\sqrt {A^{2} + B^{2} }$. In Fig. 5, the same equation is represented as tree.

Fig. 5

A typical tree structure of Pythagorean equation

Not every time this algorithm gives the same result. Some non-trivial and apparently complicated trees which finally leads to the Pythagorean theorem are:

• power(add(mul(A,A),mul(B,B)),1,2)

• power(add(mul(B,B),mul(A,A)),2,4)

• power(add(mul(B,B),mul(A,A)),3,6)

• power(add(mul(B,B),mul(A,A)),4,8)

• power(add(mul(A,A),mul(B,B)),5,10)

• power(add(mul(A,A),power(B,8,4)),1,2)

• power(add(power(A,8,4),mul(B,B)),2,4)

• power(add(power(A,4,2),mul(B,B)),5,10)

• power(add(sub(mul(A,A),sub(A,A)),mul(B,B)),1,2)

• power(add(sub(A,A),add(mul(A,A),mul(B,B))),1,2)

• power(add(mul(power(B,4,2),safe_div(A,A)),add(mul(A,A),sub(B,B))),4,8)

• power(sub(add(mul(A,A),sub(A,B)),sub(sub(A,B),mul(B,B))),5,10)

• power(add(mul(B,B),mul(A,power(power(A,4,3),3,4))),3,6)

• power(sub(mul(add(B,A),A),sub(mul(B,A),mul(B,B))),2,4)

• add(power(add(add(mul(B,B),sub(A,A)),mul(A,A)),2,4),safe_div(A,add(mul(B,B),

  mul(power(B,10,10),A))))

• power(add(add(mul(A,A),mul(B,B)),safe_div(add(B,A),power(B,5,1))),4,8)

1.1 Run Results

We have achieved 8 successful runs out of 10 test runs. The results are as follows.

• 1. power(add(mul(A, A), mul(B, B)), 1, 2)

• 2. power(add(mul(B, B), mul(A, A)), 3, 6)

• 3. power(sub(mul(B, B), sub(safe_div(power(B, 3, 5), mul(B, B)), mul(A, A))), 4, 8)

• 4. power(sub(add(mul(A, A), mul(B, B)), safe_div(sub(B, B), A)), 2, 4)

• 5. power(add(mul(A, A), add(sub(B, B), mul(B, B))), 4, 8)

• 6. power(mul(add(mul(A, A), power(B, 6, 3)), add(mul(A, A), power(B, 6, 3))), 2, 8)

• 7. power(add(mul(A, A), mul(B, B)), 4, 8)

• 8. power(add(mul(A, A), mul(B, B)), 5, 10)

All these generated trees are having fitness value near 0, i.e., 0.0407. This small error is due to choosing input points as floating number.

And the other 2 unsuccessful runs are:

• 1. add(B, safe_div(add(power(A, 5, 6), mul(A, A)), add(safe_div(mul(A, A), add(A, B)), add(B, B))))

which has fitness value of 3.795.

• 2. add(B, safe_div(safe_div(add(mul(B, A), B), safe_div(B, A)), add(add(A, B), safe_div(mul(B, B), add(B, A)))))

which has fitness value of 3.816.