A trio of Lower Sixth students took centre stage this week to deliver an intellectually rich series of presentations exploring the fascinating realms of non-Euclidean geometry, complex numbers, and higher-dimensional mathematics.
First up, Will (Queen Anne) delivered a masterful explanation of Spherical Geometry, tracing its origins through historical developments in geometry and examining how changing one of Euclid’s famous postulates—particularly the parallel postulate—leads to a completely new mathematical reality. Will vividly demonstrated how triangles on a sphere can have angles greater than 180 degrees, linking this to real-world applications such as aeroplane flight paths and satellite orbits. He even used a 3D model to bring these abstract ideas to life, before expanding into the Complex Plane and the concept of the Riemann Sphere. As a counterpoint, he introduced Hyperbolic Geometry, with its negatively curved, saddle-shaped space, and highlighted how artist M.C. Escher used it in his work. He closed with a surprising connection to spaghettification in Physics, illustrating how these mathematical ideas echo across disciplines.
Joshua (Swift) followed with an insightful journey through Complex Numbers, starting with Euler’s famous identity:
eiπ+1=0e^{iπ} + 1 = 0
Josh explored how this elegant formula allows complex numbers to be visualised as rotations, leading naturally to the concept of Quaternions. This four-dimensional extension introduces new roots of -1, known as j and k. His explanation of four-dimensional rotations (isoclinic rotations) captivated the audience, and he concluded by demonstrating a coded 4D cube in rotation, offering a rare visual insight into abstract mathematical ideas.
Finally, Beth (Feilden) brought the focus back to Spherical Geometry, delving into its astronomical applications and giving a detailed breakdown of how lines become Great Circles on the surface of a sphere. She revisited Euclid’s postulates and illustrated how these shift in different geometrical spaces. A particularly memorable moment came with her crochet-based visualisation of Hyperbolic Geometry, making the abstract beautifully tangible. Beth closed with a superbly delivered proof of the area of a triangle on a sphere, leaving the audience with a deeper appreciation of geometry’s reach across both theory and practice.
These presentations showcased not just academic excellence, but also the enthusiasm, clarity, and creativity with which our pupils explore some of the most complex ideas in modern mathematics.
