Logic & Reason
This proof technique involves assuming the negation of the statement to be proven and showing that it leads to a contradiction.
What is proof by contradiction?
This measure tells us how a surface bends in two perpendicular directions at a point.
What is scalar curvature?
This famous mathematical structure describes a one-sided surface and is often visualized as a twisted loop.
What is a Möbius strip?
This mind-bending topological construction is a ready example of a locally Euclidean space which is not second-countable.
What is the long line.
This term describes a map between sets that preserves the structure of an algebraic system.
What is a homomorphism?
This argument was extremely controversial when it was first introduced as it seemed to imply that the size of the set of real numbers between 0 and 1 was somehow larger than the set of integers.
This mathematical object generalizes the concept of curves and surfaces to higher dimensions and is studied using charts and atlases.
What is a manifold?
This infinitely repeating geometric figure is self-similar and has non-integer Hausdorff dimension.
What is a fractal?
The real line and the 2-dimensional plane are more compatible than we once thought. This counterexample demonstrates that we can continuously fill the plane with a line.
What is a space-filling curve.
This term describes a function whose graph intersects a given curve at only one point and is used in calculus to approximate functions locally.
What is a tangent function (or tangent line)?
Resolved by Alan Turing in the negative in the early 20th century, this problem stipulated that there exists an algorithm which can always determine whether a computer program will stop in finite time or run forever.
What is the Halting Problem?
Among the Platonic solids, this polyhedron has the most faces.
What is an icosahedron?
The infinite tiling of the hyperbolic plane with regular polygons is often illustrated with art by this Dutch artist.
Who is M.C. Escher?
This function is continuous everywhere but nowhere differentiable.
What is the Weierstrass function.
This term refers to a set of vectors that spans a vector space and is linearly independent.
What is a basis?
Among the most important conjectures of the 20th century, this conjecture posited that the size of the set of real numbers was the "second-smallest" infinite size. Interestingly, it was shown to be independent of the ZFC+AC set theory axioms.
What is the Continuum Hypothesis?
These higher-dimensional geometric objects are homeomorphic to standard spheres, but they have are not diffeomorphic.
What is an exotic sphere?
This theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology.
What is Tychonoff's theorem.
This is an example of a set of points on the real line that is nowhere dense, yet has positive measure.
This term describes a type of ring where every nonzero element has a multiplicative inverse, commonly used in abstract algebra.
What is a field.
This theorem states that no finite set of axioms can be used to prove every true statement.
What is Gödel's incompleteness theorem?
This "remarkable" theorem by Gauss portends that the Gaussian curvature of a 2D surface is independent of its embedding in 3D space.
What is Gauss's Theorema Egregium?
This group is the largest sporadic simple group.
What is the monster group.
A one-dimensional, compact, locally path-connected, metrizable space, whos name has a U.S. State in it.
The Hawaiian earring.
This term describes a mapping between categories that preserves the structure of objects and morphisms, widely studied in category theory.
What is a functor.