I ♥ mathematics

Some days ago I noticed a brilliant bumper sticker, saying "I ♥ topology" where the heart was replaced by a topologically homeomorphic disk (●). Amused by the idea, I tried to work out some related versions for other mathematical subjects. Here they are:

  • For geometry, the obvious choice was a cardioid:

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  • I’m thinking of changing “Algebra” into “Arithmetic” here:

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  • A connect-the-dots heart for graph theory:

    image

  • Perhaps fractals aren’t really a mathematical subject on their own, but nevertheless they are too popular:

    image

  • Too bad a heart shape doesn’t have that many symmetries:

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  • This one is difficult categorizing! It’s related to Frobenius algebras, module and representation theory, topological quantum field theory… Which mathematical subject should it represent?

    image

  • A knotted heart for knot theory:

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  • I like this one, the Lorenz attractor for chaos theory:

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  • The heart-shaped Bonne projection:

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  • I hope you recognize a Venn diagram here:

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  • Hmmm, statistics, a pie chart?

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  • Finally, this is the one it all started with:

    image

What do you guys think? I’m still thinking about a heart for genuine algebra, linear algebra, number theory, combinatorics and mathematical logic. Please share any remarks, ideas, subjects?

knot is called prime if, for any decomposition as a connected sum, one of the factors is unknotted (Livingston 1993, pp. 5 and 78). A knot which is not prime is called a composite knot. It is often possible to combine two prime knots to create two different composite knots, depending on the orientation of the two. Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of prime knots.

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CANTOR SETS - BASIC SELF-SIMILARITY 
The work of Georg Ferdinand Ludwig Philipp Cantor [1845 – 1918] – German mathematician and, in the latter third of the 19th century, the inventor of set theory, now a fundamental theory in mathematics.

IMAGES
Simple Cantor Set  [explained below]
3D Cantor Set (by Oppenheimer on deviantArt)  [Cantor dust]
Cantor Set, 2004, digital c-print (by Kevin Van Aelst)

Constructing the Cantor Ternary Set
The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments, as follows.

  • Start with a line segment – for example, the line [0,1] in the top row of the first image.
  • Cut this unbroken line into three identical parts.
  • Delete the middle of the three parts.  
    This leaves what you see in the second row. 
  • Repeat:  
    Cut each of the two lines in the second row into thirds
    and then delete the middle third of each of the two lines.
    This creates the third row.
  • Repeat …  [X]

The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated.

Sources consulted included Wikipedia articles on these subjects:
Cantor sets, Georg Cantor, Topology, Set theory

The Riemannian Tonnetz (“tonal grid,” shown above) is a planar array of pitches along three simplicial axes, corresponding to the three consonant intervals. Major and minor triads are represented by triangles which tile the plane of the Tonnetz. Edge-adjacent triads share two common pitches, and so the principal transformations are expressed as minimal motion of the Tonnetz. Unlike the historical theorist for which it is named, neo-Riemannian theory typically assumes enharmonic equivalence (G# = Ab), which wraps the planar graph into a torus.

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