geometry

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Volume to Area ratio for Regular Solids
2 months ago
- The volume-to-surface-area ratio for a sphere is V/A = r/3, where r is the radius.
- The same ratio r/3 holds for all regular solids when r is the radius of the inscribed sphere.
- For a cube with edge a, r = a/2, giving volume 8r³, area 24r², and ratio r/3.
- Other regular solids (tetrahedron, octahedron, dodecahedron, icosahedron) have more complex relationships but still yield the ratio r/3.
- Proof: By forming pyramids from each face to the center, volume = fB r/3 and area = fB, leading to ratio r/3.
- The theorem generalizes to n dimensions: the ratio of n-dimensional volume to (n-1)-dimensional boundary volume is r/n.
- The derivative of volume with respect to r gives the surface area, a classic relationship.
Brute-forcing Langley’s geometry problem with field extensions
2 months ago
- Langley's Adventitious Angles is a classic geometry problem to find an angle in a specific diagram, often solved by a clever construction.
- The problem is difficult because standard angle chasing leads to equations where the unknown angle cancels out, suggesting non-uniqueness.
- The angle is uniquely determined by the geometry, but approximate methods (drawing, trigonometry) only give an approximate answer.
- The article presents a systematic method using complex numbers and polynomial algebra to compute the exact answer without clever insights.
- The method represents points as polynomials in t = e^(2πi/36) (a 10° rotation) and performs arithmetic modulo the minimal polynomial t^12 - t^6 + 1.
- All operations (addition, multiplication, division, conjugation) are done exactly using polynomial arithmetic, yielding an exact result for the final angle.
- For the original 80°-80°-20° triangle, the computed angle is exactly 30°, as verified by the polynomial calculation returning q = t^6.
- The method can handle variants (e.g., different initial angles) and still gives exact algebraic expressions, though the final angle may be irrational.
- The approach avoids trigonometry and is based on field extensions and Galois theory, making it a reusable, systematic technique.
Reproducing a geometry theorem diagram
2 months ago
- The text describes a geometry theorem illustrated by a diagram with a unit circle, diameter AB, and perpendicular line CD.
- The blue circle is inscribed in the triangle, while the green and orange circles are inscribed in a three-sided figure with two perpendicular sides and a circular arc.
- The radius of green and orange circles is found by solving a quadratic equation, based on distances to tangent lines and the origin.
- The theorem states the diameter of the blue circle equals the sum of the radii of the green and orange circles.
- Python code is provided to generate the diagram, using numpy and matplotlib.
- The author reflects on the challenge of reproducing such geometric proofs for AI models compared to humans.
Cross-stitch prep for math nerds
2 months ago
- The most challenging aspect of cross-stitch is determining the correct quantities of supplies, which can be solved using math and geometry.
- Always round up calculations to the nearest integer to avoid underestimating supplies that could force compromises or restarts.
- To frame a pattern, first pad it to a square with side length n (the longer side) and center it; then create a circle of diameter n inside the square, scaling the square by a factor between 1 and 1/√2 if needed.
- Stitch count c (stitches per inch) determines detail and size; halving the count quarters both the actual size and thread needed.
- The minimum hoop diameter is calculated as d_min = (n√2)/c, where n is the pattern's side length in stitches.
- The theoretical minimum thread length for n stitches of the same color is (1/c)(4n-1) inches, but always buy extra to account for tension and knots.

geometry

Volume to Area ratio for Regular Solids

Brute-forcing Langley’s geometry problem with field extensions

Reproducing a geometry theorem diagram

Cross-stitch prep for math nerds