The hottest Geometry Substack posts right now

And their main takeaways
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Top Science Topics

Rubik’s abstract polytopes

A Piece of the Pi: mathematics explained • 18 implied HN points • 03 Mar 25
🔬 Science Mathematics Topology Geometry Puzzles Education
  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.

Turning a triangle into a square

A Piece of the Pi: mathematics explained • 115 implied HN points • 11 Jan 25
🔬 Science Mathematics Geometry Puzzles Research
  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.

The zigzag theorem

Infinitely More • 33 implied HN points • 04 Jan 25
🚌 Education Mathematics Geometry Learning Teaching
  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.

O-Day - Evening

Remote View • 275 implied HN points • 02 Apr 23
🔬 Science Physics Geometry Electromagnetism
  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.
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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 AI Geometry Machine Learning
  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.

Hexagonal knot mosaics

A Piece of the Pi: mathematics explained • 24 implied HN points • 03 Nov 24
🔬 Science Mathematics Geometry Topology Research
  1. Hexagonal knot mosaics are a way to represent knots on a hexagonal board. You can use different types of tiles to create them.
  2. There are three categories of hexagonal mosaics: standard, semi-enhanced, and enhanced. Each type has different rules about how crossings can occur.
  3. Research has shown the maximum number of crossings you can achieve in these mosaics. Enhanced mosaics can have the most crossings, while standard ones allow the least.

10. On Emergence

Insight Axis • 79 implied HN points • 15 May 23
🔬 Science Philosophy Systems theory Emergence Computation Geometry Cybernetics
  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 Mathematics
  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

Geometry that feels like magic

Eternal Sunshine of the Stochastic Mind • 59 implied HN points • 12 Jul 23
🔬 Science Geometry Mathematics
  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 Mathematics Geometry Physics
  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 Mathematics Teaching Geometry Learning
  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 Shapes Proofs
  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 Mathematics Geometry Physics
  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.

Corridor numbers, Fibonacci numbers, and Pascal’s triangle

A Piece of the Pi: mathematics explained • 12 implied HN points • 25 Feb 24
🔬 Science Mathematics Geometry Chess Induction
  1. Corridor numbers count ways to take diagonal steps down a corridor with fixed width. The numbers in each box form Fibonacci numbers when summed vertically.
  2. Fibonacci sequence is generated by summing the previous two terms. In the context of corridor numbers, Fibonacci numbers represent different routes to specific boxes.
  3. Pascal's triangle has rows starting and ending with 1, where each entry is the sum of two nearest entries from the row above. Circular Pascal arrays relate to corridor numbers and can produce Fibonacci numbers when subtracting specific entries.

Tips for new community members

Quantum Formalism • 39 implied HN points • 09 Mar 22
🚌 Education Mathematics Physics Geometry Topology Group Theory
  1. Start with the 'Foundation Module' YouTube playlist for basics on finite-dimensional Hilbert spaces and quantum mechanics postulates
  2. Consider auditing crash courses on topics like Topology & Differential Geometry for Lie Groups and Group Theory for advanced knowledge
  3. Exploring topics like smooth manifolds and Group Theory can be valuable not just in quantum computation but also in applied fields like ML and Cryptography

Influential Mathematicians: Sophus Lie

Quantum Formalism • 19 implied HN points • 13 Aug 20
🔬 Science Mathematics Quantum Mechanics Symmetry Geometry Algebra
  1. Sophus Lie was a Norwegian mathematician who made significant contributions to mathematics, developing the theory of continuous transformation groups that later led to Lie groups and Lie algebras.
  2. Lie Groups and Lie Algebras, named after Sophus Lie, are essential in the Hilbert space formalism of quantum mechanics, specifically in understanding symmetry and operators in quantum systems.
  3. Although Sophus Lie did not directly contribute to quantum formalism, his mathematical work has had a profound influence on areas of mathematics that are crucial to understanding quantum mechanics.

Exegesis: The Paintings of Hilma af Klint

Superb Owl • 2 HN points • 20 Jan 24
🎨 Art & Illustration Abstract art Spirituality Gender Geometry Transcendence
  1. Hilma af Klint was a pioneer of Western abstract art, drawing inspiration from Spiritualism and Theosophy.
  2. Her work was overlooked for decades but gained recognition through exhibitions drawing attention to her influence on prominent artists like Kandinsky.
  3. Gender, spirituality, and the hidden worlds of science intertwined in af Klint's paintings, conveying messages of growth and transcendence.

Tessellating the Globe

Bram’s Thoughts • 0 implied HN points • 18 Jan 24
🔬 Science Geometry Topology
  1. The world jigsaw puzzle design has imperfections when trying to tessellate it without visible seams.
  2. Two geometries proposed to fix the issue involve dividing the globe into hemispheres and deforming them into squares or into triangles corresponding to a tetrahedron's faces.
  3. Alignment of these geometries with the equator and land bodies is an interesting challenge.

Latest Community updates

Quantum Formalism • 0 implied HN points • 12 Apr 21
🚌 Education Mathematics Physics Topology Geometry
  1. The Lie Theory prerequisite mini-series will focus on point-set topology, metric spaces, and basics of differentiable manifolds.
  2. Reviewing basics of set theory, including intersections, unions, Cartesian products, and maps between sets, is recommended for the upcoming lectures.
  3. While the Lie Theory module may not be sufficient to understand Eric Weinstein's 'Geometric Unity' paper, it provides a foundational knowledge base that can ease the understanding of complex topics in differential geometry and topology.