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Mathematics | Geometry | Visual Reasoning

About Geometric Bites

Geometric Bites is a mathematics blog devoted to revealing the structure, logic and visual beauty behind mathematical ideas. It brings together rigorous reasoning, geometric intuition and visual design to show not only what mathematics does, but why it works.

Mathematics becomes more powerful when its structure is visible.

Geometric Bites develops ideas from first principles, preserves intermediate reasoning and uses geometry, algebra and visual structure to make mathematical relationships easier to understand.

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What Geometric Bites Explores

The archive covers a broad range of mathematics, from classical geometry to vectors, matrices, algebra, calculus and mathematical foundations. The emphasis throughout is on understanding structure, reconstructing results and seeing how mathematical ideas connect.

Geometry

Geometry and Constructions

Geometric proofs, coordinate methods, constructions, transformations, circles, triangles and relationships between shapes.

Vectors

Vector Geometry and Spatial Reasoning

Vectors, dot products, cross products, lines, planes, distances, projections and geometric reasoning in two and three dimensions.

Linear Algebra

Matrices and Transformations

Matrix structure, linear transformations, inverses, transposes, orthogonality, determinants, eigenvalues and their geometric interpretations.

Algebra

Algebra and Mathematical Structure

Algebraic laws, logarithms, quadratics, sequences, functions, number systems and the structural rules behind symbolic mathematics.

Analysis

Calculus, Series and Numerical Methods

Differentiation, continuity, sequences, series, induction, numerical approximation and derivations developed from first principles.

Foundations

Proof, Logic and Mathematical Foundations

Definitions, axioms, lemmas, propositions, theorems, sets, functions, groups and the logical language used to construct rigorous mathematics.

The central idea: mathematics should not be presented merely as a collection of formulas to memorise. Geometric Bites aims to reveal how results are constructed, how ideas connect and why the mathematics works.
Tiago Hands, author of Geometric Bites
About the Author

Tiago Hands

I have been creating mathematics content online since 2014, including geometric art, visual demonstrations and educational videos exploring the structure of mathematical space.

Over time, this work developed into a broader study of geometry, vectors, matrices, transformations, proof and mathematical logic. Geometric Bites provides a place to develop those ideas in greater depth and preserve the reasoning behind them.

Find more of my work
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The Purpose of Geometric Bites

Geometric Bites is intended to become more than a collection of isolated explanations. Its purpose is to build a connected body of mathematical reasoning that can be revisited, developed and expanded over time.

Build from first principles

Show where formulas, constructions and mathematical relationships come from rather than presenting them only as finished results.

Make structure visible

Use geometry, algebra and diagrams to expose relationships that can be difficult to see in symbolic mathematics alone.

Preserve the reasoning

Include intermediate steps and logical connections that are often omitted when mathematics is reduced to a formula or short procedure.

Connect mathematics and visual design

Explore how precise mathematical rules can generate elegant geometric forms and meaningful visual representations.

Support deeper understanding

Encourage readers to reconstruct ideas, test relationships and understand why techniques work instead of relying only on memorisation.

Build a lasting mathematical archive

Create an expanding library of proofs, derivations, explanations, constructions and mathematical observations that remain easy to revisit.

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A Home for Long-Form Mathematical Thinking

Social media and video are useful for presenting individual ideas, constructions and visual demonstrations. Geometric Bites serves a different purpose: it provides the space required for complete arguments, detailed derivations and explanations that can be studied at the reader's own pace.

This makes it possible to preserve the full chain of reasoning behind an idea rather than showing only the final result.

The result is an evolving mathematical archive in which geometry, algebra, vectors, matrices, calculus, proof and visual reasoning can be explored as parts of a connected subject rather than as isolated techniques.

Thank You for Visiting

Geometric Bites continues to grow through new articles, derivations, diagrams, constructions and mathematical insights. The aim is to build a long-term resource for anyone interested in understanding the clarity, structure and beauty of mathematics.

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A Geometric Way to Visualise sin(x + y) and cos(x + y)

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The angle–addition identities for sine and cosine often appear as algebraic formulas, but they can also be understood by combining two right triangles in a simple geometric construction. The calculations for the side lengths follow directly from the definitions of sine and cosine. sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y − sin x sin y Start with a right triangle of angle y and hypotenuse 1. From basic trigonometry, its horizontal and vertical sides are: cos y and sin y. Next, attach a second right triangle with angle x . Its hypotenuse is the side of length cos y from the first triangle, so its adjacent and opposite sides become: adjacent = cos x · cos y opposite = sin x · cos y Likewise, if the first triangle's vertical side sin y is used as a hypotenuse in a similar way, it contributes: adjacent = cos x · sin y opposite = sin x · sin y When the horizontal components are combined, they give the expression for cos(x + y...
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The Method of Differences — A Clean Proof of the Sum of Cubes

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The Shortest Distance Between Two Skew Lines in ℝ³

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