50 Mathematical Ideas You Really Need to Know

But more is true. In any triangle ABC the centres G, H and O, respectively the centroid, orthocentre, and circumcentre, themselv ...

Napoleon’s theorem The essential data that determines a triangle consists of knowing the length of one side and two angles. By u ...

Building with triangles The triangle is indispensable in building. Its use and strength relies on the fact that made it indispen ...

the condensed idea Three sides of a story ...

22 Curves It’s easy to draw a curve. Artists do it all the time; architects lay out a sweep of new buildings in the curve of a c ...

The conic sections Classical curves The first family in the realm of the classical curves are what we call ‘conic sections’. Mem ...

The parabola Not so obvious is the point which moves so that its distance from a point (the focus F) is the same as its perpendi ...

The logarithmic spiral Jacob Bernoulli of the famed mathematical clan from Switzerland was so enamoured with the logarithmic spi ...

Three-bar motion One aspect of 19th-century research on curves was on those curves that were generated by mechanical rods. This ...

full classification has never been carried out. The study of curves as algebraic equations is not the whole story. Many curves s ...

are ‘simple’ (do not cross themselves) and ‘closed’ (have no beginning or end). Jordan’s celebrated theorem has meaning. It stat ...

23 Topology Topology is the branch of geometry that deals with the properties of surfaces and general shapes but is unconcerned ...

all the faces are the same regular shape, all the angles between edges meeting at a vertex are equal. Tetrahedron Cube Octahed ...

Dodecahedron Icosahedron Truncated icosahedron Topology is a relatively new subject, but it can still be traced back to the Gree ...

Archimedean solids which are semi-regular. Examples can be generated from the Platonic solids. If we slice off (truncate) some c ...

A topologist might regard the donut and the coffee cup as identical but what sort of surface is different from the donut? A cand ...

Klein bottle The idea of a one-sided surface seems far-fetched. Nevertheless, a famous one was discovered by the German mathemat ...

extra dimension. Poincaré conjectured that certain closed 3-manifolds which had all the algebraic hallmarks of being three-dimen ...

24 Dimension Leonardo da Vinci wrote in his notebook: ‘The science of painting begins with the point, then comes the line, the p ...

four dimensions and in even higher n-dimensional mathematics. Many philosophers and mathematicians began to ask whether higher d ...