About

About Geometric Bites

Geometric Bites is a blog devoted to the beauty, structure, and logic of geometry. Its purpose is to present clear explanations, elegant proofs, and high-precision geometric constructions that reveal how mathematical ideas fit together. This site brings together rigorous mathematics and visual design, showing how geometry can be both exact and artistic.

What This Blog Offers

Geometric Bites explores a wide range of mathematical ideas, including:

  • geometric proofs and constructions
  • vector methods and linear algebra
  • coordinate geometry and transformations
  • projection, perspective, and spatial reasoning
  • mathematical structure explained visually and intuitively
  • geometric art generated from precise mathematical rules

Every article is written to be clear, accurate, and visually intuitive. Whether you are a student, an educator, an artist, or simply someone who enjoys the elegance of mathematics, this blog aims to sharpen understanding and inspire curiosity.

About the Author

The Author of Geometric Bites

My name is Tiago Hands. Since 2014, I have been creating mathematics content online: geometric art, visual demonstrations, and educational videos that explore the structure of mathematical space. Over the years, my work has grown into a study of geometry, vectors, transformations, and the language of mathematical logic.

You can find more of my work here:

These platforms highlight geometric constructions, proofs, animations, and visual explorations. Geometric Bites serves as the long-form home for detailed explanations, deeper insights, and complete mathematical articles that cannot fit into short-form social media posts.

The Purpose of Geometric Bites

The goal of this blog is to:

  • make geometry accessible, intuitive, and visually compelling
  • demonstrate how constructions arise from first principles
  • explain concepts with clarity, precision, and visual structure
  • connect mathematics with artistic representation
  • build a library of high-quality articles on geometry and mathematical reasoning

Each post is crafted to reveal the underlying ideas behind the mathematics, making the concepts both understandable and inspiring.

Thank You for Visiting

Thank you for exploring Geometric Bites. New articles, diagrams, constructions, and geometric insights are added regularly as the blog continues to grow. This site is designed to be a long-term resource for anyone interested in the clarity, structure, and beauty of geometry.

Popular posts from this blog

The Method of Differences — A Clean Proof of the Sum of Cubes

The Method of Differences — A Clean Proof of the Sum of Cubes The method of differences is a remarkably elegant tool for evaluating finite sums. When each term of a series can be written in the form f(r+1) − f(r) , the sum “collapses” — all interior terms cancel, leaving only a boundary expression. This behaviour is called a telescoping sum . 1) Telescoping Sums Assume the general term u r can be written as: u r = f(r+1) − f(r). Then the finite sum from r = 1 to r = n becomes: Σ u r = Σ ( f(r+1) − f(r) ). To see what happens, write out a few terms: u₁ = f(2) − f(1) u₂ = f(3) − f(2) u₃ = f(4) − f(3) ⋮ uₙ = f(n+1) − f(n) When these are added, everything cancels except the first and last pieces: Σ u r = f(n+1) − f(1). This is the essence of the method: interior structure disappears, leaving just the difference between the final and initial states. 2) A Classic Application — The Sum of Cubes We will use this technique to prove the well-known ...
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2×2 Orthogonal Matrix Mastery — A Generalised Construction

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2×2 Orthogonal Matrix Mastery — A Generalised Construction Orthogonal matrices in two dimensions reveal one of the cleanest structures in linear algebra. A 2×2 matrix is orthogonal when its columns (and rows) satisfy two conditions: They are perpendicular (their dot product is zero); They have unit length (their magnitude is one). This article presents a clear generalisation: any pair of perpendicular vectors with equal magnitude can be normalised to form an orthogonal matrix. 1. Begin with two perpendicular vectors Let the first vector be: (a, b) A perpendicular vector can be chosen as: (−b, a) Their dot products confirm orthogonality: (a, b) · (−b, a) = −ab + ab = 0 (a, −b) · (b, a) = ab − ab = 0 2. Compute their shared magnitude Both vectors have the same length: |(a, b)| = √(a² + b²) We can therefore normalise each one by dividing by √(a² + b²). 3. Form the matrix using the normalised vectors Place the two normalised vectors...
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The Maclaurin Series — A Clean Derivation

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The Maclaurin Series — A Clean Derivation Many smooth functions can be written as an infinite polynomial. When this expansion is centred at x = 0 , we obtain the Maclaurin series . This article derives the Maclaurin formula directly from repeated differentiation, showing precisely why the coefficients involve derivatives and factorials. 1) Begin with a General Power Series Suppose a function f(x) can be expressed as f(x) = a₀ + a₁x + a₂x² + a₃x³ + … + aᵣxʳ + … The constants aᵣ are real coefficients whose values we wish to determine. 2) Evaluate at x = 0 f(0) = a₀ so a₀ = f(0). 3) Differentiate Once f′(x) = a₁ + 2a₂x + 3a₃x² + … + r·aᵣxʳ⁻¹ + … Setting x = 0 eliminates all higher powers: f′(0) = a₁. Thus, a₁ = f′(0). 4) Differentiate Again f″(x) = 2·1·a₂ + 3·2·a₃x + … + r(r−1)aᵣxʳ⁻² + … Evaluate at x = 0 : f″(0) = 2! · a₂ Hence a₂ = f″(0) / 2!. 5) The General Pattern Differentiate repeatedly. After r differentiations, a...
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