Intellect PlusGeometry
527 cards. Each one is a 40-second story with a five-minute read behind it. Page 9 of 9.
A mug and a doughnut count as the same to topologyTopology tracks properties that survive continuous stretching, twisting, and bending—so long as you neither cut nor glue. Under that lens a coffee mug matches a doughnut: both are…
Polar maps locate points by radius and swing from a polePolar coordinates name a plane point by distance from a pole and angle from a polar axis—radius and azimuth, with the pole playing Cartesian origin. Spirals and central motion lov…
Polygons are closed chains of edges meeting at cornersA polygon is a plane figure whose line segments join into a closed polygonal chain. Sides are edges; meetings are vertices; an n-gon has n sides. Greek polus plus gonia means many…
Polyhedra are 3D shapes with flat faces and sharp edgesCall a solid polyhedral when flat polygonal panels meet along straight hinges at sharp corners—Greek “many bases.” Rival definitions quarrel at the edges, yet convex cases agree: …
Babylonians used Pythagoras’s rule a millennium earlyThe Pythagorean theorem says that in a right triangle the square on the hypotenuse equals the sum of squares on the other two sides—a² + b² = c². Named for Pythagoras, born around…
Simple quadrilaterals’ angles always total 360 degreesA quadrilateral is a four-sided polygon—four edges, four vertices—from Latin quadri and latus, also called a tetragon in Greek style. Figures may be simple or self-intersecting co…
Riemann made geometry’s measuring stick vary point by pointRiemannian geometry studies Riemannian manifolds—spaces where distance is length of curves on the space itself, generalising curved surfaces. Bernhard Riemann sketched the vision …
Gauss proved curvature can be felt from inside a surfaceA Riemannian manifold carries smoothly varying inner products on tangent spaces so distance, angle, length, volume, and curvature make sense. Euclidean space, spheres, hyperbolic …
Euclid made the right angle his universal angle yardstickA right angle measures exactly 90 degrees or π/2 radians—a quarter turn. If a ray meets a line so adjacent angles match, both are right. Latin angulus rectus means upright; Greek …- Geometry’s shape ignores size, place, and mirror flipsShape sketches an object’s form or outline, apart from colour, texture, or material. In geometry it also strips away position, size, orientation, and chirality—move, enlarge, rota…
- Maths calls the shell a sphere and the solid a ballA sphere is the set of points at equal distance r from a centre in three-dimensional space—the surface analogue of a circle. Radius names both the segment and its length; a diamet…
- Euclid’s point has no part yet builds every figureIn geometry a point idealises exact position with no size—a zero-dimensional primitive. Euclid called it “that which has no part,” fixed by axioms rather than smaller pieces. Anal…
Swap the circle for a hyperbola and trigonometry gets a twinPlot cosine against sine and you trace a unit circle; plot cosh against sinh and you trace the right half of a unit hyperbola. These hyperbolic functions mimic trigonometry closel…
Lord Kelvin's idea that atoms were knots spurred the first knot tablesIn the 1860s Lord Kelvin proposed that atoms were knots tied in the aether. That theory pushed Peter Guthrie Tait to build the first tables classifying knots, and the mathematics …- A metre stick is not a metre long to everyone, says relativityLength seems like the most solid quantity there is, the thing a ruler measures. Yet Einstein's special relativity says a rod one metre long in its own frame measures differently f…
- Euclid defined the line, then never used his definition againEuclid's Elements calls a straight line a breadthless length that lies evenly with its own points. Those words lean on the reader's physical intuition and undefined terms, and the…
- No single map can cover a whole circle without tearing itA manifold is a space that looks like ordinary flat space if you zoom in close enough on any point. Even the humble circle shows the catch: every patch can be flattened onto a str…
- A seismologist and a pilot measure the distance between two places differentlyAsk how far apart two points on Earth are and the answer depends on who is asking. Pilots and shipping lines want the shortest route over the surface, while seismologists care abo…
- Saccheri discovered hyperbolic geometry in 1733 while trying to disprove itGiovanni Girolamo Saccheri set out to rescue Euclid from every flaw. Assuming the parallel postulate false, he derived result after result about a strange geometry, then declared …
Topology can say two points are near without ever measuring distanceOpen sets began as a generalisation of open intervals on the number line, the stretch between two numbers without its endpoints. Topologists realised that a well-chosen collection…
Newton skipped a parabolic mirror for his 1668 telescope because it was too hard to makeA parabolic mirror sends every ray parallel to its axis to one focus, which is why satellite dishes and most modern reflecting telescopes use that shape. Descartes, Mersenne and J…- Strip a plane down to topology and it becomes an infinite rubber sheetA mathematical plane is an endless flat surface in two dimensions, but mathematicians keep changing how much structure it carries. Remove distances and straightness and it becomes…
Perelman turned down a million dollars for proving the Poincaré conjectureIn 2010 the Clay Mathematics Institute offered Grigori Perelman US$1 million for settling a question Henri Poincaré posed in 1904. He refused, saying Richard Hamilton, whose Ricci…- An ellipse keeps a constant sum to two fociFor every point on an ellipse, the distances to two fixed points called foci add up to the same total, and a circle is simply the case where the foci merge. Its area has a tidy fo…
- Euclid's axioms ruled geometry for two millenniaEuclidean geometry is the system Euclid laid out in his textbook, the Elements: accept a few intuitive postulates, then prove everything else from them. Many results were known ea…
- The Euclidean plane needs two numbers per pointA Euclidean plane is Euclidean space in two dimensions, where two real numbers pin down every point. As an affine space it has parallel lines, and its distance function adds circl…
Euler's V−E+F equals two for every convex polyhedronThe Euler characteristic is a number describing a shape's structure that stays the same however the shape is bent. For a polyhedron's surface it is vertices minus edges plus faces…
Fractals keep detail no matter how far you zoomA fractal is a shape with intricate structure at every scale, however closely you look, and its fractal dimension usually exceeds its ordinary one. Benoît Mandelbrot coined the wo…
Point-set topology rests on continuity, compact, connectedGeneral topology, also called point-set topology, supplies the basic set-theoretic definitions that differential, geometric and algebraic topology all build on. Its three central …
Geometry began as land measure in Mesopotamia and EgyptGeometry studies the distance, shape, size and relative position of figures, and alongside arithmetic it is one of mathematics' oldest branches. Its earliest records come from Mes…
Homology turns shapes into sequences of abelian groupsHomology began in algebraic topology as a way to turn a shape's holes into algebra. At its core is a chain complex, a string of abelian groups linked by maps whose back-to-back co…
Homotopy asks if one map can melt into anotherTwo continuous maps between spaces are homotopic if one can be smoothly morphed into the other, and the morph itself is called a homotopy. The word joins Greek homós, same, and tó…- Hyperbolas are the two-branched mirror conicA hyperbola is a smooth plane curve made of two mirror-image branches, like a pair of endless bows. It appears when a plane slices both halves of a double cone without touching th…
- A circle is every plane point at one fixed distance from a centerA circle is every point in a plane lying at one fixed distance from a centre, an idea people knew before recorded history. The full moon and a sliced fruit show it in nature. The …
Compactness makes infinite spaces behave like finite setsIn topology, compactness is the infinite analogue of a finite list: you can always thin a sprawling open cover down to finitely many sets and still blanket the space. Maurice Fréc…- Slice a cone and you get ellipses, parabolas, or hyperbolasA conic section is the curve where a flat plane slices a double cone. Tilt the plane one way and you get a closed ellipse, with the circle as a special case; set it parallel to th…
A set is convex if it keeps every segment between its pointsPick any two points in a convex set and the straight segment joining them never leaves it. A solid cube passes that test, while anything hollow or dented, such as a crescent, fail…- Coordinates turn geometry into ordered lists of numbersA coordinate system pins down each point of a space with one or more numbers, turning geometry problems into arithmetic and back again, the foundation of analytic geometry. The si…
A cube packs six equal squares into twelve equal edgesThe cube is a polyhedron with eight vertices, twelve equal edges and six square faces, a special case of the cuboid, parallelepiped and rhombohedron. Plato's Timaeus paired it wit…
Modern curves are continuous images of intervals—sometimes wild onesEuclid called a line a breadthless length; curved lines were the flexible kin of straight ones. Today a curve is often the continuous image of an interval in a space. That definit…
Differential geometry gave Einstein the language of curved spacetimeDifferential geometry studies smooth shapes and spaces, called manifolds, using vector calculus and linear and multilinear algebra. It grew from ancient measurements of a spherica…- Dimension counts how many coordinates a point needsInformally, a space’s dimension is the fewest coordinates required to fix any point—one for a line, two for a plane, three for ordinary space. Spacetime’s four dimensions fuse eve…
- Euclidean distance is the straight shortcut physics usually meansDistance measures how far apart things are—physical length, a rough “two counties over,” or social and psychological separation in soft sciences. In classical physics the default …
- Analytic geometry turns shapes into equations on axesAlso called coordinate or Cartesian geometry, the field studies figures with a coordinate system instead of purely synthetic constructions. It underpins algebraic, differential, d…
- Angles measure openings between rays that share a vertexTwo lines meeting at a point form an angle whose sides are those lines and whose vertex is the shared point. The same word names both the figure and its size. Measures link to cir…
- Area measures how much room a region covers on a surfacePlane area sizes a flat shape; surface area sizes the boundary of a solid. Intuitively it is how much thin material or one coat of paint would cover the figure—the two-dimensional…
- Cartesian axes pin every point with signed distancesIn the plane each point gets a unique pair of real coordinates—signed distances to two fixed perpendicular oriented axes meeting at the origin (0, 0). Three mutually perpendicular…