Geometric Bites

Arithmetic and Geometric Sequences Arithmetic and geometric sequences are two fundamental types of numerical progressions. They describe how quantities grow or shrink by addition or by multiplication, and they form the foundation for topics such as series, summation formulas, and exponential growth. 1. Arithmetic Sequence An arithmetic sequence is a list of numbers in which each term differs from the previous one by a fixed amount called the common difference d . a, a + d, a + 2d, a + 3d, … , a + (n − 1)d a – first term d – common difference The n th term, denoted T n , is given by: T n = a + (n − 1)d Each new term is obtained by adding d to the previous term. The difference between consecutive terms remains constant: T k+1 − T k = d Example: If a = 4 and d = 3, the sequence is 4, 7, 10, 13, 16, … The 20th term is T 20 = 4 + (20 − 1)×3 = 61. 2. Geometric Sequence A geometric sequence is a list of numbers where each term is fo...

What Is an Isomorphism? In mathematics, an isomorphism is a function that shows two mathematical objects have the same structure. Although the objects may look different, an isomorphism demonstrates that they behave in exactly the same way with respect to the operations that define them. If such a map exists, the objects are called isomorphic . An isomorphism tells us that two systems are essentially the same, differing only by a relabelling of their elements. The Basic Idea An isomorphism is a function: f : A → B that must be: Injective — different elements of A map to different elements of B. Surjective — every element of B comes from some element of A. Together these mean f is bijective , and so it has an inverse: f⁻¹ : B → A No information is lost moving from A to B or back. Preserving Structure Bijectivity alone is not enough. An isomorphism must also preserve structure. For groups, this means: f(a ⋆ b) = f(a) ∘ f(b) for all a, b ∈ A ,...

Proof by Induction: The Sum of Squares Formula Theorem. For any integer n ≥ 1, ∑ i=1 n i² = (1/6)·n·(n+1)·(2n+1) Proof (by mathematical induction) 1. Basis Step For n = 1: LHS = 1² = 1 RHS = (1/6)·1·(1+1)·(2·1+1) = (1/6)·1·2·3 = 1 Therefore, the formula holds for n = 1. 2. Inductive Hypothesis Assume the statement is true for some integer k ≥ 1: ∑(i=1→k) i² = (1/6)·k·(k+1)·(2k+1) 3. Inductive Step We must show it is true for n = k + 1: ∑(i=1→k+1) i² = [∑(i=1→k) i²] + (k+1)² = (1/6)·k·(k+1)·(2k+1) + (k+1)² = (1/6)(k+1)[k(2k+1) + 6(k+1)] = (1/6)(k+1)[2k² + 7k + 6] = (1/6)(k+1)(k+2)(2k+3) This matches the same form with k replaced by k+1: (1/6)·(k+1)·((k+1)+1)·(2(k+1)+1) Hence, the formula holds for n = k + 1. 4. Conclusion Since the statement is true for n = 1 (basis step), and true for n = k + 1 whenever it is true for n = k (inductive step), it follows by the principle of mathematical induction that ∑(i=1→n) i² = (1/6)·n·(n+1)·(2n+1) for al...

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...

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 ...

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...

Barycentric Coordinates Made Clear: From a UV Triangle to a 3D Triangle This post explains, in plain language, why a simple triangular region in a 2-dimensional parameter space can describe every point of a real triangle in 3-dimensional space. Through one clean affine formula, the 2D parameters (u, v) determine points on a 3D triangle. Once this connection is seen, concepts such as interpolation, texture mapping, geometric modelling, and FEM become much easier to understand. Two Worlds Connected by a Map 1) Parameter World (UV-space, 2D) Start in a flat coordinate plane labelled by two parameters, u and v . Consider the square defined by 0 ≤ u ≤ 1 0 ≤ v ≤ 1 If we cut this square along the diagonal line u + v = 1 (or equivalently v = −u + 1 ), we keep only the triangular region 0 ≤ u ≤ 1 0 ≤ v ≤ 1 u + v ≤ 1 This triangle has vertices (0,0) , (1,0) , (0,1) and is called the standard 2-simplex . Every point inside it is a convex mixture of its three corners. 2...

What Are Barycentric Coordinates? Barycentric coordinates provide a way to describe any point inside a triangle using the triangle’s own vertices as a reference. Instead of relying on the usual x–y axes, we express a point as a weighted combination of the three corners. The Core Idea Let a triangle have vertices A , B and C . Any point P inside (or on) the triangle can be written as P = αA + βB + γC The numbers α, β and γ are the barycentric coordinates of P. They indicate how strongly each vertex contributes to P. For this expression to make geometric sense, the three weights must satisfy α + β + γ = 1 This condition ensures that P behaves like a weighted average — a “blend” of A, B and C — rather than drifting away from the triangle. As long as all three values are non-negative ( α, β, γ ≥ 0 ), the point lies somewhere within the triangle. Examples α = 1 , β = γ = 0  →  P = A α = β = 0.5 , γ = 0  →  midpoint of AB α = ...

Injective, Surjective, and Bijective Functions In mathematics, a function describes how elements of one set are assigned to elements of another. Three important properties capture how completely and uniquely a function connects its domain to its codomain: injective , surjective , and bijective . These properties tell us whether different inputs can share the same output and whether every possible output is used. Injective (One-to-One) A function is injective if different inputs always produce different outputs. No two distinct elements in the domain are allowed to map to the same result in the codomain. Formally, a function f : A → B is injective if: f(a₁) = f(a₂) ⇒ a₁ = a₂ Equivalently, if a₁ ≠ a₂ , then f(a₁) ≠ f(a₂) . Example: f(x) = e x from ℝ → ℝ is injective. Different inputs produce different outputs, but not every real number is reached, so it is not surjective. Surjective (Onto) A function is surjective if every element of the codomain is reach...

What Is a Group in Mathematics? In abstract algebra, a group is a set G together with a binary operation (written as * ). The pair (G, *) is called a group when the operation satisfies the four conditions below. Closure: for all g₁, g₂ ∈ G , the product g₁ * g₂ is still in G . Identity: there exists an element e ∈ G such that for all g ∈ G , e * g = g * e = g . This element e is called the identity. Inverses: for each g ∈ G there exists an element g⁻¹ ∈ G such that g * g⁻¹ = g⁻¹ * g = e . Associativity: for all g₁, g₂, g₃ ∈ G , g₁ * (g₂ * g₃) = (g₁ * g₂) * g₃ . These four conditions are exactly what is required for (G, *) to be a group. Side note: Commutativity An operation is commutative if swapping the elements does not change the result: a * b = b * a . Commutativity is not required for a group to exist. It is important not to confuse commutativity with associativity . These are distinct i...

Essential Elements of Algebra Problem Solving Solving equations in algebra is about preserving truth while working step by step toward the unknown. Although equations can look complicated, most of the time we are simply applying a small set of rules with care. If these rules are followed, the solution you reach will be valid; if they are broken, the result becomes unreliable. This post introduces three essential principles that underpin almost all algebraic manipulation. Follow them consistently, and you will have a strong foundation for solving equations with confidence. 1) Start with a True Statement Everything in algebra begins with a statement that is already true. If the starting point is false, no amount of manipulation can produce a trustworthy conclusion. When you write an equation such as: 2x + 3 = 13 you are asserting that doubling a number and then adding three gives thirteen. This is your initial truth. All further steps must preserve this truth. If y...

Function Composition: A Simple Way to Organise Your Mathematics Function composition is one of the most useful tools in mathematics. It allows us to combine several steps into a single process, keeping our work neat, organised, and easy to reuse. Rather than performing one operation after another by hand, composition lets us build those steps into a single function. Once you become comfortable with function composition, you never want to go back to doing everything manually. It reduces clutter, helps you work systematically, and allows you to achieve remarkable results with only a few lines of equations. What Is Function Composition? A function takes an input, performs an operation, and produces an output. Function composition takes this idea further: it links functions together so that the output of one becomes the input of the next. We write this as: (f ∘ g)(x) = f(g(x)) This means that g acts first, then f . The circle symbol ∘ simply means “do one funct...

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...

Continuity and Differentiability (Clear, Compact Guide) We study functions of a single real parameter u . A scalar function is written as φ(u) , and a vector function as R(u) = (x(u), y(u), z(u)) . All the ideas below are based on ordinary one-variable limits. 1) What it means for a function to be continuous at u (scalar case) A function φ is said to be continuous at u if its value changes smoothly as u changes slightly. Formally, for every ε > 0 , there exists a δ > 0 such that |φ(u + Δu) − φ(u)| < ε whenever |Δu| < δ . 2) Continuity of a vector function Let R(u) = (x(u), y(u), z(u)) . The function R is continuous at u if each component x(u) , y(u) , and z(u) is continuous at that same point. Equivalently, using any fixed norm |·| on ℝ³ : for every ε > 0 , there exists a δ > 0 such that |R(u + Δu) − R(u)| < ε whenever |Δu| < δ . 3) Differentiability (first order) A scalar or vector function is differentiable at u if the limit (F(u + Δ...

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...