Visualising - Advanced
Proofs with Pictures
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Vector Journeys
Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?
Vanishing Point
How can visual patterns be used to prove sums of series?
When the Angles of a Triangle Don't Add up to 180 Degrees
This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.
3D Treasure HuntLive
Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?
Tetra Square
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
Stonehenge
Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
Coordinated Crystals
Explore the lattice and vector structure of this crystal.
Escriptions
For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
Classical Means
Use the diagram to investigate the classical Pythagorean means.
Middle Man
Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?
Classic Cube
The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?
Mach Attack
Have you got the Mach knack? Discover the mathematics behind exceeding the sound barrier.
Set Square
A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?
A Rolling Disc - Periodic Motion
Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?
Wrapping Gifts
A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?
Fitting Flat Shapes
How efficiently can various flat shapes be fitted together?
Five Circuits, Seven Spins
A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.
Circles Ad Infinitum
A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?
Maximum Scattering
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
Ford Circles
Can you find the link between these beautiful circle patterns and Farey Sequences?
Cheese Cutting
In this problem we see how many pieces we can cut a cube of cheese into using a limited number of slices. How many pieces will you be able to make?