The hottest Geometry Substack posts right now

And their main takeaways
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“Lights Out” on a triangular grid

A Piece of the Pi: mathematics explained • 30 implied HN points • 22 Mar 26
🔬 Science Geometry
  1. The triangular Lights Out game reduces to linear algebra over the field with two elements: pressing a button toggles bits mod 2, pressing a button twice cancels, order doesn’t matter, and any solution is a subset of buttons pressed once.
  2. Solvability and uniqueness depend on the kernel of the toggle map: if the kernel is only the empty set (ℓ=0) then every starting state has a unique solution, which occurs for certain side lengths such as 1, 3, 4, 7, 8, 9, 11, 15, 16, 17, 20, and 21.
  3. If the kernel is nontrivial (ℓ>0) there are nonzero button patterns that have no effect and some starting configurations cannot be solved; the kernel is a 2^ℓ-sized vector space over GF(2) and its patterns often form visually striking shapes like the Sierpiński triangle.

You gotta think outside the hypercube

lcamtuf’s thing • 5101 implied HN points • 26 Jan 26
🕹 Technology Geometry
  1. You can build a tesseract wireframe by extending the same edge-construction rules from a square to a cube and then to 4D, which yields 16 vertices and 32 edges.
  2. Rotations in four dimensions are still planar operations that act on pairs of axes, so animations come from applying familiar 2D rotation formulas to axis pairs like XZ or Z🌀.
  3. There are many ways to project 4D to 2D with different tradeoffs—cavalier, cabinet, isometric, perspective, and fisheye—and a mixed approach (isometric for XYZ plus perspective or fisheye for the fourth axis) gives the clearest, most informative views.

What's the deal with Euler's identity?

lcamtuf’s thing • 5917 implied HN points • 08 Nov 25
🔬 Science Geometry
  1. Euler's identity, which is e^(iπ) + 1 = 0, connects five important math constants: e, π, 0, 1, and i. It shows how complex numbers and trigonometry blend together in a fascinating way.
  2. The number i is known as the imaginary unit, and it allows us to represent two-dimensional rotations. When we multiply by i, it represents a 90° turn in the complex plane.
  3. Using Euler's formula, we can relate complex exponentials to trigonometric functions. This connection helps us understand circular motion in a mathematical way.

Surfaces and sequence graphs

A Piece of the Pi: mathematics explained • 66 implied HN points • 31 Jan 26
🔬 Science Geometry
  1. You can build a graph by placing n vertices in a cycle and linking them according to the rank order of the first n terms of a real sequence, and as n grows these sequence graphs reveal striking geometric patterns.
  2. Graphs coming from the Kronecker sequence (multiples of the golden ratio mod 1) can be drawn on a torus without crossings, typically after removing the edge from n−1 to 0.
  3. Graphs from the van der Corput sequence embed into the Chamanara surface — a highly singular, infinite‑handle (“Loch Ness monster”) surface made by identifying shrinking boundary segments of a square — and finite approximations avoid the worst singularities so they can be visualized.
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Modular curve stitching

A Piece of the Pi: mathematics explained • 66 implied HN points • 10 Dec 25
🔬 Science Geometry
  1. Connecting each point k to a·k (mod m) on an m-point circle produces lacy modular stitch patterns that, as m grows, converge to smooth cycloid curves; for positive a these limit curves are epicycloids with a−1 petals.
  2. The same curves arise from a continuous 'dancing planets' model where two bodies orbit an origin an integer number of times and a tether between them is sampled; reversing a planet's direction turns epicycloids into hypocycloids.
  3. Sampling can cause aliasing so different orbital parameters can produce identical sampled patterns when the sample size m equals |αδ−βγ|, which explains why seemingly different stitch parameters sometimes look the same.

Colouring identical cuboids

A Piece of the Pi: mathematics explained • 30 implied HN points • 30 Dec 25
🔬 Science Geometry
  1. The number of colours needed depends on the cuboids' sizes and orientations; even with all pieces aligned, some stacks need five colours (2×2×1), others need four (3×1×1) or three (2×1×1).
  2. Letting identical cuboids meet at right angles in the same plane can raise the colour count — for example, 2×1×1 blocks in mixed planar orientations can require five colours.
  3. Allowing arbitrary orientations makes the problem harder: some constructions with 4×1×1 cuboids force at least six colours, there is a proven upper bound of 12 in that case, and it’s unknown whether six is the true maximum overall.

O-Day - Evening

Remote View • 275 implied HN points • 02 Apr 23
🔬 Science Geometry
  1. The O-Day - Evening post discusses the electromagnetic properties of the Great Pyramid.
  2. The post delves into the connections between alchemy, sacred geometry, and the 'Great Work'.
  3. There are references to scientific articles and historical figures within the context of the post.

Edge 370: A Deep Dive Into AlphaGeometry: Google DeepMind’s New Model that Solves Geometry Problems Like a Math Olympiad Gold-Medalist

TheSequence • 364 implied HN points • 15 Feb 24
🕹 Technology Geometry
  1. Google DeepMind has created AlphaGeometry, an AI model that can solve complex geometry problems at the level of a Math Olympiad gold medalist using a unique combination of neural language modeling and symbolic deduction.
  2. The International Mathematical Olympiad announced a $10 million prize for an AI model that can perform at a gold medal level in the competition, which historically has been challenging even for top mathematicians.
  3. Geometry, as one of the difficult aspects of the competition, traditionally requiring both visual and mathematical skills, is now being tackled effectively by AI models like AlphaGeometry.

Turning a triangle into a square

A Piece of the Pi: mathematics explained • 115 implied HN points • 11 Jan 25
🔬 Science Geometry
  1. Henry Dudeney showed in 1902 that you can cut an equilateral triangle into four pieces and rearrange them into a square with the same area. This is a fun example of how shapes can transform while keeping their total area the same.
  2. The Wallace–Bolyai–Gerwien theorem explains how you can rearrange two shapes with the same area into each other through cutting, but Dudeney's method is unique because the pieces stay connected during the transformation.
  3. Recent research proved that you can't turn a triangle into a square using fewer than four pieces without flipping any. This shows how specific and tricky these geometric dissections can be.

10. On Emergence

Insight Axis • 79 implied HN points • 15 May 23
🔬 Science Geometry
  1. Emergence occurs when an entity has properties that its individual parts do not possess, displaying behaviors that only emerge in interaction.
  2. Simple computational or geometric rules can lead to unpredictable and complex outputs, showcasing the beauty of emergence.
  3. Emergence, as seen in cybernetics with Braitenberg's Vehicles, demonstrates how simple structures can give rise to emergent, complex behavior, hinting at the potential for understanding the universe through simple rules.

Antiprisms

Pershmail • 78 implied HN points • 24 Apr 23
🔬 Science Geometry
  1. Antiprisms are shapes made by connecting triangles to the edges of bases, twisting one base to fit them together
  2. An antiprism has specific properties like vertices, edges, and faces, which can be calculated using Euler's polyhedron formula
  3. Square antiprisms, or 'squaps', have intriguing features like cross-sections of octagons and can be understood with geometry toys

Games in projective space

A Piece of the Pi: mathematics explained • 48 implied HN points • 16 Jun 25
🔬 Science Geometry
  1. Projective geometry removes the concepts of distance and parallel lines, which changes how we think about shapes and space. It's a unique way to understand geometry differently.
  2. In projective space, there are still points, lines, and planes, but the rules are different from traditional geometry. This can lead to interesting and complex interactions.
  3. Games can be explored in the context of projective space, allowing for creative new strategies and outcomes based on its unique properties.

The curvature of space

Infinitely More • 33 implied HN points • 27 Jul 25
📖 Philosophy Geometry
  1. Spherical geometry has positive curvature, which means circles on a sphere are smaller than expected compared to flat surfaces.
  2. In hyperbolic space, there are way more locations nearby than in regular space, making it easier to get lost or have many places to explore.
  3. Although spherical and hyperbolic geometries are quite different, they can seem similar to a person at a small scale, just like how our everyday experience seems like flat geometry.

Geometry that feels like magic

Eternal Sunshine of the Stochastic Mind • 59 implied HN points • 12 Jul 23
🔬 Science Geometry
  1. In geometry, certain geometric properties can hold true regardless of how the figures are drawn, leading to aesthetically pleasing and eternal truths.
  2. Specific theorems like Morley's trisector theorem and Napoleon's theorem showcase the magic of geometry by revealing surprising relationships within triangles.
  3. Concepts like Simson's line and Țițeica's 3 circles theorem demonstrate the beauty and elegance of geometry, inspiring us to appreciate the world through the lens of mathematics.

Double Bubbles

Pershmail • 58 implied HN points • 14 Apr 23
🔬 Science Geometry
  1. Double bubbles minimize surface area by using interesting film connections.
  2. For fencing in different areas with minimal material, the double bubble shape is ideal.
  3. The standard double bubble minimizes perimeter between two areas.

Shape of the Week: Zonogon

Pershmail • 58 implied HN points • 17 Mar 23
🚌 Education Geometry
  1. The 'Shape of the Week' feature introduces a new geometric shape each week to expand knowledge and make learning fun.
  2. A zonogon is a parallelogram with point symmetry and can be dissected into multiple parallelograms, creating an interesting mathematical pattern.
  3. Regular zonogons can produce beautiful dissections, and studying them can lead to exploring concepts like Minkowski's First Theorem.

Salinons are the Shape of the Week

Pershmail • 58 implied HN points • 30 Mar 23
🔬 Science Geometry
  1. Salinons are a geometric shape with a unique construction involving four semicircles.
  2. Salinons have an interesting area formula that relates to a circle with the same area.
  3. You can replace the semicircles in a salinon with other shapes, like rectangles or triangles, and still maintain certain area relationships.

Sphericons

Pershmail • 39 implied HN points • 28 Apr 23
🔬 Science Geometry
  1. The sphericon is a shape that wobbles when twisted, and it's made of two pieces resembling bicones.
  2. The sphericon has square dimensions and a 90 degree angle from one end extending down.
  3. Generalizations of the sphericon, called polycons, roll in a wobbly way and include shapes like hexacons, octacons, and decacons.

Tiling the plane with congruent polygons

A Piece of the Pi: mathematics explained • 18 implied HN points • 29 Jun 25
🔬 Science Geometry
  1. You can't cover a flat surface with regular pentagons because their angles don't fit together perfectly. The angle of a pentagon is 108°, and it's not a number that evenly divides into 360°.
  2. However, there are other shapes, like certain hexagons and quadrilaterals, that can tile the plane without any gaps. These shapes can fit together nicely to fill space.
  3. Tiling is a fun way to explore patterns and geometry, showing how shapes can interact in creative and mathematical ways. It leads to interesting discoveries in both art and mathematics.

Boomerang tilings

Infinitely More • 15 implied HN points • 25 Jun 25
🔬 Science Geometry
  1. Boomerangs are special shapes called nonconvex quadrilaterals. They can be used to explore interesting questions about tiling.
  2. The main question is whether a convex polygon can be tiled completely using just a few boomerangs. This is a challenging mathematical problem.
  3. Finding a solution to this problem requires careful thought and may not be easy. Just because one attempt fails, it doesn’t mean that it can’t be done at all.

The zigzag theorem

Infinitely More • 33 implied HN points • 04 Jan 25
🚌 Education Geometry
  1. The zigzag theorem states that when you create a zigzag pattern in a rectangle, the triangles formed below this pattern take up exactly half the area of the rectangle.
  2. Even if the zigzag lines sometimes move backward without crossing, the triangles will still cover half the rectangle's area due to how the bases and heights of the triangles are calculated.
  3. This theorem is interesting because it holds true even if the zigzag involves an infinite number of lines.

Rubik’s abstract polytopes

A Piece of the Pi: mathematics explained • 18 implied HN points • 03 Mar 25
🔬 Science Geometry
  1. Rubik's Cube can be made in different shapes, like a tetrahedron or dodecahedron, instead of just the classic cube. These variations have their own names, like the Megaminx for the dodecahedron.
  2. A new study explains how to think about these puzzles in more dimensions, counting how many different ways they can be turned or rearranged. This includes understanding shapes like the hypercube.
  3. The math behind a Rubik's Cube shows interesting patterns, like counting sticks and pieces, and this can help us understand all kinds of shapes and designs better.