The School of Numbers & Logic · mathematics, pure and applied
Geometry & Trigonometry
Why shapes behave as they do, and how to prove it — from congruent triangles to the trigonometry that measures the unreachable.
Classical plane geometry with proof at the center — the course where mathematics first shows its whole method.
Syllabus · 5 units · ~32 hours
Unit I — Foundations
Points, lines, planes, and the language of definition · Angles and angle relationships · What an axiom is and why geometry starts from them · First proofs: vertical angles
Unit II — Triangles & Congruence
Triangle congruence criteria: SSS, SAS, ASA · Writing a two-column and a paragraph proof · Isosceles triangles and their theorems · The triangle inequality
Unit III — Parallels & Polygons
Parallel lines and transversals · The angle sum of a triangle and of any polygon · Quadrilaterals and their hierarchy · Similarity and scale
Unit IV — Circles
Chords, arcs, and central angles · Inscribed angles and their theorem · Tangent lines · Circumference and arc length
Unit V — Area & Pythagoras
Area formulas derived, not issued · The Pythagorean theorem, proved twice · Solid figures: surface area and volume
From right-triangle ratios to the unit circle and periodic functions — measuring heights, distances, and anything that oscillates.
Syllabus · 4 units · ~26 hours
Unit I — Right-Triangle Trigonometry
Sine, cosine, tangent as ratios · Solving right triangles · Angles of elevation and depression · Measuring a tree, a cliff, a building
Unit II — The Unit Circle
Radian measure and why mathematicians prefer it · Extending sine and cosine to all angles · Reference angles and exact values · The graphs of sine and cosine
Unit III — Identities
The Pythagorean identity · Sum and difference formulas · Double-angle formulas · Verifying identities without circular reasoning
Unit IV — Any Triangle at All
The law of sines and its ambiguous case · The law of cosines · Triangulation: how surveyors and astronomers use it
Descartes' bargain — trade figures for equations — carried through lines, circles, and the three curves cut from a cone.
Syllabus · 3 units · ~18 hours
Unit I — Geometry with Coordinates
Distance, midpoint, and slope as tools of proof · Proving classical theorems by coordinates · Choosing coordinates wisely
Unit II — Circles & Parabolas
The circle from its definition · The parabola as locus: focus and directrix · Completing the square to read an equation's shape
Unit III — Ellipses & Hyperbolas
The ellipse and its two foci · The hyperbola and its asymptotes · Eccentricity: one dial for all four curves · Orbits and reflectors: conics at work
The mathematics of pattern — rigid motions, wallpaper groups, and the five solids the Greeks could not add to.
Syllabus · 4 units · ~12 hours
Unit I — Rigid Motions
Reflections, rotations, translations, glide reflections · Composing motions · The symmetry type of a figure
Unit II — Patterns in a Strip and on a Wall
The seven frieze patterns · The seventeen wallpaper groups, surveyed · Reading the symmetry of real ornament — tiles, textiles, brickwork
Unit III — Tilings
Which regular polygons tile the plane · Semiregular tilings · Aperiodic tilings and the Penrose surprise
Unit IV — Polyhedra
The five Platonic solids and why there are no more · Euler's formula: vertices minus edges plus faces · Archimedean solids and the geometry of a soccer ball
Geometry with distance removed — what survives stretching, and why a coffee cup and a doughnut count as the same shape.
Syllabus · 4 units · ~40 hours
Unit I — Topological Spaces
Open sets: nearness without distance · Topologies, bases, and examples · Closed sets, closure, and boundary
Unit II — Continuity
Continuous maps in topological language · Homeomorphism: when two spaces are the same · The coffee cup and the doughnut, made precise
Unit III — Compactness & Connectedness
Connectedness and the intermediate value theorem · Compactness and why it earns its keep · The Heine–Borel theorem
Unit IV — Surfaces
The Möbius band and orientability · The torus and the Klein bottle · The Euler characteristic as an invariant · The classification of surfaces, stated and believed