Things don't need to be perfectly round to count as balls. Footballs, golf balls, and tennis balls have seams and dimples, and even the Earth isn't perfectly round because it has mountain ranges and is slightly flattened at the poles. Yet, we still think of all of these things as spherical.
The only way to transform something with no holes (such as the ball in the top left of the image) to something with one or two holes is to tear or puncture the object. This figure first appeared on Plus in the article The shape of things to come.
In mathematics, this tolerance of deformation is captured by the field of topology. Topology considers two objects the same if you can deform one into the other without tearing or cutting: only bending, stretching and squeezing is allowed. The most famous example is that, topologically speaking, a ball is the same as a bowl, and a donut the same as a coffee cup. If a ball or a donut were made out of clay you could bend and stretch them to form a bowl or a coffee cup respectively. But you can't turn a bowl into a coffee cup without tearing or cutting a hole.
This suggests that, when it comes to surfaces such as spheres or donuts, the number of holes is what makes the difference between two objects. And this is indeed correct. Mathematicians have proved that many surfaces that naturally spring to mind are topologically equivalent to either a sphere (with zero holes), the surface of a donut (technically called a torus, which has one hole), or a torus with two holes, three holes, etc. The number of holes is what defines such a surface in the topological sense.
A sphere, a torus, a surface with two holes, and a surface with three holes. Image of sphere: Geek3, used under Creative Commons licence.
By "most surfaces that naturally spring to mind" we mean surfaces that are closed and orientable. Closed just means that the surface has no boundary you could fall off if you were walking around on it (an example of a surface with a boundary is a disc) and that you can paint it with a finite amount of paint (technically, that it is compact). Orientable means, loosely speaking, that there's a clear difference between the two sides of the surface (eg an inside and an outside). That's in contrast to the famous Möbius strip, which only has one side. (See here for more detail on a surface being closed and orientable.)
Coming back to ordinary objects, we now see that a bowl belongs in the zero hole category, a cup into the one hole category, a pair of spectacles (without the glasses) into the two hole category, and a pretzel into the three hole category. In technical terms, the number of holes of a surface is called its genus.
Surfaces are two-dimensional objects, but mathematicians are also able to think in higher dimensions. A natural question that arises here is how to topologically characterise a three-dimensional sphere (which nobody can properly visualise, so don't worry if you can't). The question proved very difficult to answer. It involves one of the most famous problems in maths — the Poincaré conjecture — a $1,000,000 prize and prestigious medal, and a reclusive mathematician who would accept neither. Find out more in this article.
In higher dimensions, surprisingly, things get a little easier again. The nature of so-called manifolds — analogues of surfaces — is well understood for these dimensions.
This article is based in part on the Plus article The shape of things to come. To find out more about topology you can also read:
FAQs
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
What is the formula for topology? ›
Combinatorial Topology.
This is the number (V - E + F), where V, E, and F are the number of vertices, edges, and faces of an object. For example, a tetrahedron and a cube are topologically equivalent to a sphere, and any “triangulation” of a sphere will have an Euler characteristic of 2.
What are the math topology rules? ›
A topology rule can monitor spatial relationships of features in a single feature class, or the relationships that exist between feature classes. Only simple feature classes in the same dataset can participate in a topology (Annotation, dimension, and geometric network features are not simple features).
What is min() in math? ›
min() method returns the number with the lowest value.
Is there calculus in topology? ›
Topology and analysis (calculus) are rather distinct branches of mathematics. But in general it works the other direction. Analysis relies on properties of the real line, complex field, and multidimensional Euclidean space which depend on principles that are a part (a small part) of general (point-set) topology.
What are 5 basic topological rules? ›
Use your features' spatial relationships and behavior to define topology rules
- Parcels cannot overlap. ...
- Stream lines cannot overlap and must connect to one another at their endpoints.
- Adjacent counties have shared edges. ...
- Adjacent Census Blocks have shared edges. ...
- Road centerlines must connect at their endpoints.
Theorem 1. Tychonoff's theorem: An arbitrary product of compact space is compact in the product topology. With Tychonoff's theorem, we can get some interesting results. For example, the topological space r0, 1sr0,1s with product topology is compact as a product of compact space r0, 1s.
What is topology in layman's terms? ›
topology, branch of mathematics, sometimes referred to as “rubber sheet geometry,” in which two objects are considered equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching, and shrinking while disallowing tearing apart or gluing together parts.
What is the formula for min in math? ›
The second way to find the minimum value comes when you have the equation y = ax^2 + bx + c. If your equation is in the form y = ax^2 + bx + c, you can find the minimum by using the equation min = c - b^2/4a. The first step is to determine whether your equation gives a maximum or minimum.
What is the min strategy in math? ›
an arithmetic strategy in which children faced with an addition problem start with the largest addend and count up from there. For example, for the problem, 3 + 2 = ?, a child would say “3… 4, 5.”
math. min() is an inbuilt method in Java which is used to return Minimum or Lowest value from the given two arguments. The arguments are taken in int, float, double and long.
What math is required for topology? ›
Applied Topology
The prerequisite is linear algebra together with an introduction to proofs. However, a more advanced version of this course could be taught with a prerequisite of a one-semester course in undergraduate analysis.
Is topology part of algebra? ›
As mathematical disciplines, Algebra and Topology are different theories. But there are many interplay applications of Algebra to Topology and wise verse, such as hom*ology theory.
Is linear algebra used in topology? ›
Based on classical results from algebraic topology techniques, we show that a (co)chain complex and all associated combinatorial operations are readily represented using standard techniques from linear algebra, giving rise to a Linear Algebraic Representation (LAR) scheme.
Is topology part of geometry? ›
Topology is almost the most basic form of geometry there is. It is used in nearly all branches of mathematics in one form or another.
