One point on the sphere for every point of the plane
stereographic is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
At its defaults
show: "circles"
show: "conformal"
show: "area"
show: "atlas"
show: "mobius"
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the image of 0 lies on the circle ×14
- the ray to 0 passes through the pole ×14
- the image of -2 lies on the circle ×13
- the ray to -2 passes through the pole ×13
- a circle missing the pole projects to a circle, to within a fitted residual ×1
- a circle through the pole projects to a straight line ×1
- and no two of them meet ×1
- and the south chart does have one there ×1
- at least two finite points are carried across the projection ×1
- between 2 and 24 rays are drawn — one ray is not a correspondence ×1
- between a hundred and four thousand points checked ×1
- between three and seven fibres, each at a latitude strictly inside the sphere ×1
- dim and rays are read by the ray views alone ×1
- each fibre lies on the unit three-sphere ×1
- each transformation is normalised to determinant one ×1
- every patch has the same area on the sphere ×1
- every two fibres are linked exactly once, counted from the drawn curves ×1
- nearly every sample lay in the overlap ×1
- none of the circles in this list passes through the pole ×1
- only the parabolic map has a repeated fixed point ×1
- the crossing angle is the same on the sphere and in the plane ×1
- the fitted transformation is invertible ×1
- the image is on the ray from the pole through the point ×1
- the image lands on the plane ×1
- the north chart has no value at its own pole ×1
- the projected patches are wildly unequal even though the originals are equal ×1
- the projection is drawn from a circle or from a sphere ×1
- the rotation, read in the plane, is exactly the fitted Möbius transformation ×1
- the three chosen points determine the transformation ×1
- the trace classifies the transformation as the label claims ×1
- the two charts differ by the reciprocal at every point of the overlap ×1
- the two charts miss different points ×1
- the view is one the family draws ×1
- there are pairs to link ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Angles survive and areas do not
Stereographic projection takes every circle on the sphere to a circle or a line, and every crossing angle to itself. It does both exactly, with no approximation anywhere, and it destroys area so thoroughly that a patch near the pole can be a thousand times its neighbour's size.
NumberEvery triple, on one circle
Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.
TopologyOne chart is never enough
Stereographic projection matches the sphere minus a point with the whole plane, and the missing point is not a blemish to be tidied away. It is a theorem — no single flat picture covers a sphere — and the repair is two pictures with a rule for passing between them.
TopologyA sphere is a plane plus one point
Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.
TopologyThe circles that fill a three-sphere
A three-sphere is filled by circles — one through every point, no two meeting, every two linked exactly once. Stereographic projection is the only way anybody sees it, and the projected picture is a nest of circles on tori whose linking can be counted off the drawing.
GeometryThe map that trades circles for lines
Send every point to the one on the same ray whose distance multiplies with it to a fixed number, and circles become lines, lines become circles, angles survive untouched, and a ring of tangent circles falls out of a ring of equal ones.
GeometryThe number four points agree on
One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.
TopologyThe sphere that complex numbers live on
Add one point to the complex plane and it becomes a sphere. The rotations of that sphere are exactly the maps written as one linear expression divided by another, so a fact about turning a ball is a fact about dividing polynomials.
GeometryThe triangle that a globe gets wrong
On a sphere, a right triangle with legs of fifty and sixty degrees has a hypotenuse of seventy-two, not seventy-eight. The theorem is not approximately true there — it is false, and what replaces it says exactly how much room the surface has.
TopologyTwo pieces, in every dimension
A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.
TopologyWhere the fixed point escapes
The theorem asks for a set that is closed, bounded and free of holes. Drop any one of the three and a map appears that moves every single point — and in each case the point that should have stayed still can be seen leaving.