@TopologyFact

Tweets on topology and geometry from @JohnDCook.

JohnDCook.com
Joined January 2010
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"I want to start a new field of geometry where two things are equivalent if you can stretch one into the other without tearing." "I've got just what you need: a collection of sets closed under finite intersections and arbitrary unions." "Huh?!"
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A hyperbolic knot is a knot that has a complement that can be given a complete metric of constant curvature -1.
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A satellite knot can be formed by first taking a knot inside an unknotted torus, then making a second “companion” knot of that torus. The satellite knot lives in the “orbit” of the companion knot.
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A torus knot can be drawn on the surface of an unknotted torus.
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A box in a ball in a box, in high dimension johndcook.com/blog/2021/04/1…

Box in ball in box in high dimension

The n-ball fits better in the n-cube than the n-cube fits in the n-ball if and only if n is less than or equal to 8.

johndcook.com
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The acceleration of geodesic curve has no tangential component. It has no acceleration from the perspective of the surface it is on.
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Kawasaki paper folding theorem en.wikipedia.org/wiki/Kawasa…

Kawasaki's theorem - Wikipedia

en.wikipedia.org
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William Thurston proved that all knots fall into one of three categories: torus, satellite, and hyperbolic.
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Insane Properties of the Closed Long Ray and the Long Line serokell.io/blog/long-line-p…

Insane Properties of the Closed Long Ray and the Long Line

Long rays and long lines have some very interesting (even strange) properties. Read this article to brush up your topology skills.

serokell.io
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The product of compact sets is compact.
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A topological model for provability logic johndcook.com/blog/2026/10/0…

A topological model for provability logic

Provability logic, a.k.a. Gödel-Löb logic, is a modal logic in which the box operator means a proposition is provable. It has a different topological model.

johndcook.com
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There is no retraction of a disk to its boundary, i.e. no continuous function from the disk to its boundary that leaves the boundary fixed.
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Visualizing Hyperbolic Honeycombs hyperbolichoneycombs.org/
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Topological models of modal logic johndcook.com/blog/2026/10/0…

Topological models of modal logic

Topological spaces provide models of modal logics where the box operator is interpreted as the interior of a set and the diamond operator its closure.

johndcook.com
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Total curvature of a knot: johndcook.com/blog/2019/10/0…

Total curvature of a torus knot

The Fary-Milnor theorem says the total curvature of a knot must exceed 4 pi. We demonstrate this with examples.

johndcook.com
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'In most physical applications of tensor calculus, the space is Riemannian ... and we can suit our convenience as to whether we express that tensor in contravariant, covariant, or mixed form.' – Synge & Schild
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On simplicial complexes, singular and simplicial homology are isomorphic.
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The covariant derivative measures the rate of change of a vector field along a curve as seen from a surface.
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Radius, circumference, and area in non-Euclidean geometry johndcook.com/blog/2022/08/2…

Radius, circumference, and area in non-Euclidean geometry

In non-Euclidean geometry, the circumference and area of a circle are different functions of the radius.

johndcook.com
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Exotic spheres: There are 28 smooth structures on the 7-sphere. (Milnor and Kervaire)
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List of fractals by Hausdorff dimension en.wikipedia.org/wiki/List_o…

List of fractals by Hausdorff dimension - Wikipedia

en.wikipedia.org
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