Inversion in a circle of radius 1
inversion 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.
With nothing chosen
An Apollonian gasket, 125 circles in
Eight circles touching three
What inversion does to circles and to lines
Inversion keeps the angle between two curves
A ring of 6 circles touching two others and each other
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 circles round vertex 0 subtend exactly a full turn at it ×7
- ring 1: and the circle it surrounds from outside ×2
- ring 1: and touches the containing circle from inside ×2
- ring 1: each circle touches the next ×2
- a circle missing the centre inverts to a circle ×1
- a circle orthogonal to the mirror is carried to itself ×1
- a circle through the centre inverts to a line ×1
- a point inside goes outside and a point outside comes in ×1
- a point on the mirror circle is left exactly where it is ×1
- and a line missing the centre inverts to a circle ×1
- and a point of the circle lands on it ×1
- and conjugates the imaginary part ×1
- and every edge of the triangulation is a tangency of two circles ×1
- and every point of the straight line lands on its circle ×1
- and its curvature is a whole number ×1
- and one for each of the eight ways of choosing inside or outside ×1
- and so do their images, because a circle's image is a circle ×1
- and the two short gaps overshoot ×1
- and touches each of them on the page ×1
- between two and four points are inverted, none of them at the centre ×1
- both images are circles here ×1
- centre and all ×1
- circles whose vertices are not joined do not overlap ×1
- each drawn circle really touches each given one ×1
- each image gap is the original gap divided by the two distances from the centre ×1
- each new circle satisfies Descartes's relation with its three parents ×1
- every curvature in the packing is a whole number ×1
- every pair of the seed circles touches ×1
- every point of the circle lands on its image ×1
- every point of the circle through the centre lands on its image ×1
- every walk round an interior vertex closes on the circles already placed ×1
- four points on one circle have a real cross-ratio ×1
- four points, in order round the circle and none of them nearly coincident ×1
- in both parts ×1
- neither boundary passes through the inversion centre ×1
- no chain circle passes through the inversion centre ×1
- no point being inverted sits at a mirror's centre ×1
- no point being inverted sits at the centre ×1
- off the circle the identity becomes a strict inequality ×1
- off the circle the three images are not collinear ×1
- on a circle the products satisfy Ptolemy's identity exactly ×1
- one inversion leaves the real part of the cross-ratio alone ×1
- so the two short image gaps add to the long one ×1
- the angle between the two curves is the same after the map as before ×1
- the chord's end sits on the mirror ×1
- the circle's image is a circle ×1
- the concentric ring closes exactly when the radius ratio is the sine ×1
- the constructed point is the one inside the mirror ×1
- the drawn ring is genuinely lopsided rather than a disguised concentric one ×1
- the first mirror has a sensible radius ×1
- the four points are not concyclic, so their cross-ratio has a real imaginary part to conjugate ×1
- the fourth point is pushed off the circle by less than half a radius ×1
- the gasket is drawn to between one and six generations ×1
- the image of the centre is not the centre of the image ×1
- the image of the crossing point is on both image circles ×1
- the inner circle has a sensible radius ×1
- the inversion centre is inside the inner circle and off its centre ×1
- the line inverted does not pass through the centre ×1
- the mirror circle has a sensible radius ×1
- the ring has between three and twelve circles ×1
- the ring round the centre closes up ×1
- the second mirror is offset by a sensible amount ×1
- the seed satisfies Descartes's relation ×1
- the tangent construction lands on the same point the formula gives ×1
- the three centres are not collinear ×1
- the three given circles lie outside one another ×1
- the three images fall on one straight line ×1
- the two circles really cross ×1
- the two distances from the centre multiply to the square of the radius ×1
- the view is one the family draws ×1
- three boundary radii, repeated round the outside ×1
- three circles in general position admit exactly eight tangent circles ×1
- three given circles, each with a sensible radius ×1
- two inversions put the cross-ratio back exactly ×1
- which passes through the centre of the mirror ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
Curvatures that stay whole
Four circles touching one another satisfy an equation in their curvatures. Read it as a quadratic and the second solution is the first subtracted from something — so a packing that starts with whole numbers stays whole forever.
GeometryEight circles touching three
Draw three circles. How many circles touch all three? The answer is eight, the count is a fact about signs rather than about geometry, and the classical way to find them is to move the problem somewhere it becomes easy.
GeometryEvery flat graph is a pile of circles
A graph that can be drawn without crossings can be drawn in one particular way: as circles, one per vertex, touching exactly when their vertices are joined. The picture is not a choice — it is determined, up to the group two inversions generate.
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.
ComputationThe straightedge buys nothing
Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.