Geometric Bites

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...

Normalised Vectors: A Clear and Intuitive Guide Vectors can have any length, but many mathematical problems only depend on direction. To separate direction from magnitude, we normalise the vector. This produces a new vector of length 1 that points the same way as the original. Normalised vectors are central to geometry, physics, 3D graphics, transformations, and any setting where orientation matters. By working with a unit vector, calculations become simpler, cleaner, and more meaningful. What Is a Normalised Vector? A normalised vector is a vector with magnitude 1. It keeps its direction but loses its original size. Some simple unit vectors include: (1, 0, 0) — magnitude 1 (0, 1, 0) — magnitude 1 (0, 0, 1) — magnitude 1 These are the standard basis vectors. In general, any non-zero vector can be transformed into a unit vector by dividing by its magnitude. Normalising a Vector in 2 Dimensions For a 2D vector (a, b) , the length is: √(a² + b²) To n...

The Dot Product Identity and the Cosine Rule in ℝ 3 In this article we derive the dot product identity A · B = |A| × |B| × cos(θ) and show how this identity leads directly to the cosine rule, using a combination of coordinate algebra and geometric interpretation. 1. Vectors in ℝ 3 Let the vectors be: A = (a 1 , a 2 , a 3 ) B = (b 1 , b 2 , b 3 ) Their difference is: A - B = (a 1 - b 1 , a 2 - b 2 , a 3 - b 3 ) The squared magnitude of this difference vector is: |A - B| 2 = (a 1 - b 1 ) 2 + (a 2 - b 2 ) 2 + (a 3 - b 3 ) 2 . 2. Expanding the Square of the Difference Expand each component: (a 1 - b 1 ) 2 = a 1 2 - 2a 1 b 1 + b 1 2 (a 2 - b 2 ) 2 = a 2 2 - 2a 2 b 2 + b 2 2 (a 3 - b 3 ) 2 = a 3 2 - 2a 3 b 3 + b 3 2 Adding these three expansions gives: |A - B| 2 = (a 1 2 + a 2 2 + a 3 2 ) + (b 1 2 + b 2 2 + b 3 2 ) - 2(a 1 b 1 + a 2 b 2 + a 3 b 3 ). Recognise the squared magnitudes: |A| 2 = a 1 2 + a 2 2 ...

Homogeneous Coordinates: A Simple and Intuitive Primer In ordinary geometry, we use familiar coordinates such as (x, y) in 2D or (x, y, z) in 3D. These work well, but they have one major limitation: not all geometric transformations fit neatly into this system—especially translations and perspective projections. To unify everything into one clean mathematical framework, we introduce homogeneous coordinates . They provide a simple way to treat every transformation—from translations to perspective projection— using only matrix multiplication. 1. Why Do We Need Something New? In ordinary coordinates: rotations are matrices, scalings are matrices, shears are matrices, translations are not matrices . Translation is the “odd one out.” This creates friction in computer graphics, robotics, and projective geometry, where we want one system that handles everything the same way. Homogeneous coordinates fix this by adding one extra coordinate. 2. The Bas...

What Is an Isometry? In geometry, some transformations distort shapes by stretching, squashing, or bending them. Others preserve the shape perfectly. These distortion-free transformations are called isometries . An isometry keeps every distance between points exactly the same. The object may move, rotate, or flip, but its size and structure remain unchanged. 1. The Core Idea A transformation f is an isometry if: distance( f(x), f(y) ) = distance( x, y ) for every pair of points x and y. This is the mathematical way of saying: nothing is stretched, compressed, or distorted. 2. A Simple Real-Life Analogy Place a phone on a desk. Slide it. Rotate it. Flip it. It remains the same phone—same size, same shape, same geometry. Each of those motions is an isometry . 3. What Isometries Never Do An isometry does not : stretch or shrink a shape, shear it diagonally, bend or curve it, change angles or proportions. It behaves exactly like movi...

The Trapezoidal Rule — A Visual, First-Principles Introduction 11 November 2025 · @mathematics.proofs To understand integration deeply, it helps to think geometrically. Instead of memorising formulas, we begin by observing shapes. The goal is simple: break the interval into small pieces, estimate the area on each piece, and add everything together. Rectangles give a basic approximation. But functions rarely behave perfectly flat on every interval. A better idea is to allow the top edge to tilt. This leads us naturally to trapezoids . 1) Partitioning the Interval Consider an interval from a to b . We divide it into n equal parts. Δx = (b − a) / n x i = a + i·Δx for i = 0, 1, 2, …, n At each x i , we record the height of the function f(x i ). These sample values will guide our area estimates. 2) One Slice: Rectangle + Triangle Focus on a single subinterval [x i , x i+1 ]. If we draw a vertical line at x i and take a height of f(x i ), we obtain a rectangle o...

Common Transformations in Geometry: A Beginner’s Guide In geometry, a transformation is a rule that changes the position or appearance of a shape or a set of points. Some transformations simply move a shape to a new location, while others may turn it, resize it, or reflect it. Understanding these ideas helps us describe movement and change in a clear mathematical way. This guide introduces the most common transformations: translation, rotation, scaling, reflection, shear, and projection. Each section includes a simple example to make the ideas easier to follow. 1) Translation — Moving A translation shifts every point of a shape by the same amount. Nothing about the shape itself changes — not its size, not its proportions, and not its orientation. Only its position is different. Example: Imagine a triangle on graph paper. If every point of the triangle moves 3 units to the right and 2 units up, the triangle looks exactly the same — it simply appears somewhere else on th...

How to Derive the Derivative of a Vector Function 🧮 Let’s start with a vector function of a single variable: R(u) = x(u)î + y(u)ĵ + z(u)k̂ Here, x , y , and z are differentiable scalar functions of a real parameter u . This means that as u changes, the point R(u) moves through space — tracing a smooth curve. ✨ The Goal We want to find dR/du — the rate at which the vector R(u) changes with respect to u . Proof By definition of the derivative: dR/du = lim Δu→0 [R(u+Δu) − R(u)] / Δu = lim Δu→0 [x(u+Δu)î + y(u+Δu)ĵ + z(u+Δu)k̂ − (x(u)î + y(u)ĵ + z(u)k̂)] / Δu = lim Δu→0 [(x(u+Δu)−x(u))/Δu]î + [(y(u+Δu)−y(u))/Δu]ĵ + [(z(u+Δu)−z(u))/Δu]k̂ = (dx/du)î + (dy/du)ĵ + (dz/du)k̂ Interpretation The derivative dR/du is itself a vector — one that points in the direction of motion of R(u) and whose magnitude gives the speed of change. Each component ( x , y , z ) behaves just like an ordinary function, so we can differentiate them individually and recombine t...

The Pythagram Defined Contributors over a 2.5 Year Period: Tiago Hands ( Final Construction ), Carlos Luna-Mota ( Egyptian Triangles ), Andrzej Kukla ( The Rhombus with Area of 3 ) 06 October 2025 Abstract The Pythagram is a planar geometric structure derived from the 3 : 4 : 5 right triangle and the orthographic projection of a cube. Its construction through defined Cartesian coordinates reveals six interrelated Pythagorean sub-figures that encode proportional symmetry across multiple dimensions. This document formalises the coordinates, connections, and geometric properties of the Pythagram as a reproducible mathematical entity. Coordinate Data Group 1 — Central Core A(16, 18), B(14, 18), C(12, 16), D(12, 14), E(14, 12), F(16, 12), G(18, 14), H(18, 16), I(15, 15), W(15, 17.5), X(12.5, 15), Y(15, 12.5), Z(17.5, 15) Group 2 — Inner Square L(15, 20), M(10, 20), N(10, 15), O(10, 10), P(15, 10), Q(20, 10), R(20...

Using a geometrical logo that can be reconstructed mathematically offers a variety of benefits, particularly in terms of design precision, versatility, and brand identity. Here's a detailed breakdown of these advantages: 1. Precision and Consistency Exact Reproduction : Mathematical definitions ensure that the logo can be reproduced exactly in any size or medium without distortion or loss of detail. Scalability : Geometrical designs based on mathematical constructs can scale seamlessly, maintaining their proportions and clarity from a tiny icon to a billboard. 2. Aesthetic Appeal Symmetry and Balance : Logos designed using geometric principles often incorporate symmetry and proportion, which are inherently pleasing to the human eye. Timelessness : Geometrical designs tend to feel clean and modern, yet timeless, making them less susceptible to trends. 3. Ease of Modification Flexible Adjustments : Since the logo's structure is based on mathematical principles, chang...

Today I was watching a video about how to construct a dodecahedron . In the first part of the clip, it was shown how to get the golden ratio with a specific type of geometric construction . I was absolutely intrigued by it and decided to figure out if the construction did indeed yield the result phi. Remarkably and beautifully, I was able to confirm the result. My workings are shown below. I live for these eureka moments! Part 1 Part 2 Part 3   Part 4

Below I demonstrate how to derive tan((3π)/8) , sin((3π)/8) and cos((3π)/8) from scratch. I use geometry and also algebra . Part 1  Part 2 Part 3 Part 4 Part 5  Part 6  If you'd like quick updates every time I post, follow me on Twitter at: https://twitter.com/tiago_hands .

Below I demonstrate how to find the line that passes through two intersecting circles. Part 1 Part 2 Part 3   There is a free interactive graph that shows why this is correct. https://www.desmos.com/calculator/z9rh2ri9bz

Recently, I produced some recursive geometric art that got a pretty good amount of attention on the internet. Here it is: So, what was the secret of its success? It was a solid geometric drawing I made beforehand. With that solid geometric drawing, I was able to see the proportions required to produce my animation. There is a clarity that geometric drawings bring because of their rigidity . As you can see in the image above, the squares get smaller . Their lengths decrease by a factor of 1/2 with each iteration . When these lengths are applied to vertical and horizontal lines what you get is pure mathematical magic... The pattern in my video. How about the equations ? Well, if I tell you everything it would be like revealing the entire story of a great movie before you've watched it. I would hate to spoil your creative experience. :-)

This proof solves a popular internet geometry problem found here: https://twitter.com/RonySarker71/status/1428469920443076609 By utilising the radius of the circle , one can arrive at the conclusion that y= √(ax). If a=1, then you get y= √(x).  

A square the quarter the size of another square will have a radius half the size. I show this in my workings below.   Why is this useful? Well, this truth can be used to create recursive square art using graphing apps like Desmos . In one of my next posts you'll see exactly what I mean. I have some art already planned. Stay tuned!