Find the Number of Triangles

  • Context: Undergrad 
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SUMMARY

The discussion focuses on calculating the number of triangles in a complex geometric figure using combinatorial methods rather than brute force counting. Participants suggest creating a 25x25 matrix to represent connections between vertices and employing a counting algorithm to identify triangles formed by intersecting lines. The conversation highlights the importance of considering symmetries and the need to account for singular triangles, which have overlapping vertices. Ultimately, the method involves calculating combinations of intersecting lines and adjusting for duplicates to arrive at an accurate count.

PREREQUISITES
  • Understanding of combinatorial mathematics
  • Familiarity with matrix representation of graphs
  • Knowledge of triangle properties in geometry
  • Experience with algorithmic counting techniques
NEXT STEPS
  • Research combinatorial geometry techniques for counting shapes
  • Learn about graph theory and its applications in geometry
  • Explore algorithms for counting intersections in geometric figures
  • Study methods for identifying and eliminating duplicates in combinatorial counts
USEFUL FOR

Mathematicians, educators, and students interested in advanced geometry, combinatorial analysis, and algorithm design will benefit from this discussion.

  • #31
Wondering if this is how Garret Lissi got his start.

{edit}
I finally counted 160....

1767352449187.webp
 
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  • #32
My artwork isn’t as pretty but here it is anyway. Each size triangle is represented only once. There are four shapes out of 22 that are left-right symmetric wrt to the whole figure are multiplied by 4 and the rest by 8.
IMG_4659.webp
 
  • #33
bob012345 said:
My artwork isn’t as pretty but here it is anyway. Each size triangle is represented only once. There are four shapes out of 22 that are left-right symmetric wrt to the whole figure are multiplied by 4 and the rest by 8.
...
At least you didn't screw up. I've just noticed multiple errors in my image. Glad I said it would take me a week to count all these stinkin' triangles!
 

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