Ars Mathematica

Inactive

My apologies, I nearly forgot I started this Tumblr. You see, I’m in my last year at university and so the past few months have been all about applying to grad to school. I think I got three solid letters of recommendations, a beautifully written personal statement, and a well rounded CV; but I honestly don’t know where exactly I fit in the spectra of math students. And don’t tell me to lurk on gradcafe because that leads to undue anxiety. Anyways, I applied to 19 schools, which is too many, but hopefully one of them sees fit to admit me. I don’t know if I’m going to keep this tumblr or just create a professional math blog. Maybe both. I have some life organizing to do after that hellstorm of applications. Anyways, happy new years. 2014 is squarefree! And only good things can come in a squarefree year! 

-Bradley.


Visualizing Projective Measured Laminations of the Five Punctured Sphere →

The above picture was generated by Dr. David Dumas of University Illinois at Chicago in order to visualize projective measured laminations of the five punctured sphere. A lamination is fairly simple, it is layering curves onto a topological space (a surface). What the adjectives “projective” and “measured” indicate are beyond me. Click through to find the above picture animated. 


isomorphismes:

Lagrangian of the Standard Model by Matilde Marcolli (via Lieven Le Bruyn)
 
This is her illustration of why higher mathematics needs to reduce the complication of something we can read but probably not understand.
group theory
quotienting
equivalence-classes
simple observations like “2+2=4 in how many different ways?”
are the kinds of tools that can reduce, reduce, simplify, reduce something like the above into what, with a minor encyclopædic knowledge, can finally be comprehensible.

 
For higher level examples let’s look at Michi Johanssons’ blog and Jacob Lurie’s undergraduate thesis.


Even if you don’t know what the symbols mean, you can see that their arrangement is simple—not sequential in the way of the Lagrangian of the standard model of particle physics, but you see pieces like ABA and BAB—simple patterns embedded in this vast encyclopædic framework, still ABA BAB is understandable by a human.

Likewise in Lurie’s example we hear about “lacing” and 27 complex variables at once—but it’s all been reduced, reduced, reduced so that the pieces can fit in a logic that a human (who knows the vocabulary) can understand through its simplicity.

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isomorphismes:

Lagrangian of the Standard Model by Matilde Marcolli (via Lieven Le Bruyn)

 

This is her illustration of why higher mathematics needs to reduce the complication of something we can read but probably not understand.

are the kinds of tools that can reduce, reduce, simplify, reduce something like the above into what, with a minor encyclopædic knowledge, can finally be comprehensible.

a family of curves / solutions to ODE's broken down by a Lie actiona rotationally symmetric family of curves / solutions to an ODE

 

For higher level examples let’s look at Michi Johanssons’ blog and Jacob Lurie’s undergraduate thesis.

\begin{diagram}  R^4 &\rTo^{\begin{pmatrix}A&0&BAB&0\\0&B&0&ABA\\0&0&A&B\end{pmatrix}} &  R^3 &\rTo^{\begin{pmatrix}A&0&BAB\\0&B&ABA\end{pmatrix}} &  R^2 &\rTo^{(A \quad B)} & R & \rTo\\  \dTo^{f_3} && \dTo^{f_2} && \dTo^{f_1} & \rdTo^{(a \quad b)\circ\epsilon} \\  R^3 &\rTo^{\begin{pmatrix}A&0&BAB\\0&B&ABA\end{pmatrix}} &  R^2 &\rTo^{\begin{pmatrix}A&B\end{pmatrix}} & R & \rTo^\epsilon &\mathbb F_2 & \rTo & 0 \end{diagram}

H^*(D_8,\mathbb F_2)=\mathbb F_2[x^1,y^1,z^2]/\langle xy\rangle

Even if you don’t know what the symbols mean, you can see that their arrangement is simple—not sequential in the way of the Lagrangian of the standard model of particle physics, but you see pieces like ABA and BAB—simple patterns embedded in this vast encyclopædic framework, still ABA BAB is understandable by a human.

the Lie algebra E6 can be represented with restricted bits of {27 complex numbers}

Likewise in Lurie’s example we hear about “lacing” and 27 complex variables at once—but it’s all been reduced, reduced, reduced so that the pieces can fit in a logic that a human (who knows the vocabulary) can understand through its simplicity.

image


I was reading  Gil Kalai’s  Combinatorics blog when I came upon this surprising post tagged “art.” These three are paintings by French artist Bernar Venet.  These painting belong to a series called “Equation Paintings” (click his name to find more paintings), and from what I can tell are an instance of artistic appropriation, since these are actually diagrams from published works of mathematics. This is exactly the type of thing that this blog has set out to do, to set images generated for the elucidation of mathematics in a context generally reserved for artistic images. 
The title of these works are as follows: 1. “Three-Dimensional Faces and Quotients of Polytopes,” 2. “Canonical Field Quantization,” and 3. “Zig-Zag Path Z_uv between Nodes u and v in a Planar Mesh.”
The only image I have found the origin of is the first, which comes from a paper that Kalai wrote (with Kleinschmidt and Meisinger): "Three Theorems, with Computer-Aided Proofs, on Three-Dimensional Faces and Quotients of Polytopes.” The image selected is only a portion of the computer generated output included in the proof of one of the papers main theorems. You can jump to Kalai’s post to read his explanation of the theorem, but I will briefly try myself. We all know what polygons are (triangles, rectangles, pentagons), and we all know what polyhedra are (cubes, tetrahedrons, icosahedrons). Well a polytope is just a higher dimensional version of this, its just mathematicians got tired of affixing different words to “poly-” so they just say “polytopes.” You can make a polyhedron with an arbitrarily large face (for example a thousand sided polygon). It turns out if a polytope has dimension greater than 8, then it must have a “small” three dimensional face. One does not expect this to be the case, but it is. This theorem (Kalai admits) has absolutely no purpose. But it is one of those great little self contained mysteries of mathematics. 
The final image is the original text from Kalai’s paper.I was reading  Gil Kalai’s  Combinatorics blog when I came upon this surprising post tagged “art.” These three are paintings by French artist Bernar Venet.  These painting belong to a series called “Equation Paintings” (click his name to find more paintings), and from what I can tell are an instance of artistic appropriation, since these are actually diagrams from published works of mathematics. This is exactly the type of thing that this blog has set out to do, to set images generated for the elucidation of mathematics in a context generally reserved for artistic images. 
The title of these works are as follows: 1. “Three-Dimensional Faces and Quotients of Polytopes,” 2. “Canonical Field Quantization,” and 3. “Zig-Zag Path Z_uv between Nodes u and v in a Planar Mesh.”
The only image I have found the origin of is the first, which comes from a paper that Kalai wrote (with Kleinschmidt and Meisinger): "Three Theorems, with Computer-Aided Proofs, on Three-Dimensional Faces and Quotients of Polytopes.” The image selected is only a portion of the computer generated output included in the proof of one of the papers main theorems. You can jump to Kalai’s post to read his explanation of the theorem, but I will briefly try myself. We all know what polygons are (triangles, rectangles, pentagons), and we all know what polyhedra are (cubes, tetrahedrons, icosahedrons). Well a polytope is just a higher dimensional version of this, its just mathematicians got tired of affixing different words to “poly-” so they just say “polytopes.” You can make a polyhedron with an arbitrarily large face (for example a thousand sided polygon). It turns out if a polytope has dimension greater than 8, then it must have a “small” three dimensional face. One does not expect this to be the case, but it is. This theorem (Kalai admits) has absolutely no purpose. But it is one of those great little self contained mysteries of mathematics. 
The final image is the original text from Kalai’s paper.I was reading  Gil Kalai’s  Combinatorics blog when I came upon this surprising post tagged “art.” These three are paintings by French artist Bernar Venet.  These painting belong to a series called “Equation Paintings” (click his name to find more paintings), and from what I can tell are an instance of artistic appropriation, since these are actually diagrams from published works of mathematics. This is exactly the type of thing that this blog has set out to do, to set images generated for the elucidation of mathematics in a context generally reserved for artistic images. 
The title of these works are as follows: 1. “Three-Dimensional Faces and Quotients of Polytopes,” 2. “Canonical Field Quantization,” and 3. “Zig-Zag Path Z_uv between Nodes u and v in a Planar Mesh.”
The only image I have found the origin of is the first, which comes from a paper that Kalai wrote (with Kleinschmidt and Meisinger): "Three Theorems, with Computer-Aided Proofs, on Three-Dimensional Faces and Quotients of Polytopes.” The image selected is only a portion of the computer generated output included in the proof of one of the papers main theorems. You can jump to Kalai’s post to read his explanation of the theorem, but I will briefly try myself. We all know what polygons are (triangles, rectangles, pentagons), and we all know what polyhedra are (cubes, tetrahedrons, icosahedrons). Well a polytope is just a higher dimensional version of this, its just mathematicians got tired of affixing different words to “poly-” so they just say “polytopes.” You can make a polyhedron with an arbitrarily large face (for example a thousand sided polygon). It turns out if a polytope has dimension greater than 8, then it must have a “small” three dimensional face. One does not expect this to be the case, but it is. This theorem (Kalai admits) has absolutely no purpose. But it is one of those great little self contained mysteries of mathematics. 
The final image is the original text from Kalai’s paper.I was reading  Gil Kalai’s  Combinatorics blog when I came upon this surprising post tagged “art.” These three are paintings by French artist Bernar Venet.  These painting belong to a series called “Equation Paintings” (click his name to find more paintings), and from what I can tell are an instance of artistic appropriation, since these are actually diagrams from published works of mathematics. This is exactly the type of thing that this blog has set out to do, to set images generated for the elucidation of mathematics in a context generally reserved for artistic images. 
The title of these works are as follows: 1. “Three-Dimensional Faces and Quotients of Polytopes,” 2. “Canonical Field Quantization,” and 3. “Zig-Zag Path Z_uv between Nodes u and v in a Planar Mesh.”
The only image I have found the origin of is the first, which comes from a paper that Kalai wrote (with Kleinschmidt and Meisinger): "Three Theorems, with Computer-Aided Proofs, on Three-Dimensional Faces and Quotients of Polytopes.” The image selected is only a portion of the computer generated output included in the proof of one of the papers main theorems. You can jump to Kalai’s post to read his explanation of the theorem, but I will briefly try myself. We all know what polygons are (triangles, rectangles, pentagons), and we all know what polyhedra are (cubes, tetrahedrons, icosahedrons). Well a polytope is just a higher dimensional version of this, its just mathematicians got tired of affixing different words to “poly-” so they just say “polytopes.” You can make a polyhedron with an arbitrarily large face (for example a thousand sided polygon). It turns out if a polytope has dimension greater than 8, then it must have a “small” three dimensional face. One does not expect this to be the case, but it is. This theorem (Kalai admits) has absolutely no purpose. But it is one of those great little self contained mysteries of mathematics. 
The final image is the original text from Kalai’s paper.

I was reading  Gil Kalai’s  Combinatorics blog when I came upon this surprising post tagged “art.” These three are paintings by French artist Bernar Venet.  These painting belong to a series called “Equation Paintings” (click his name to find more paintings), and from what I can tell are an instance of artistic appropriation, since these are actually diagrams from published works of mathematics. This is exactly the type of thing that this blog has set out to do, to set images generated for the elucidation of mathematics in a context generally reserved for artistic images. 

The title of these works are as follows: 1. “Three-Dimensional Faces and Quotients of Polytopes,” 2. “Canonical Field Quantization,” and 3. “Zig-Zag Path Z_uv between Nodes u and v in a Planar Mesh.”

The only image I have found the origin of is the first, which comes from a paper that Kalai wrote (with Kleinschmidt and Meisinger): "Three Theorems, with Computer-Aided Proofs, on Three-Dimensional Faces and Quotients of Polytopes.” The image selected is only a portion of the computer generated output included in the proof of one of the papers main theorems. You can jump to Kalai’s post to read his explanation of the theorem, but I will briefly try myself. We all know what polygons are (triangles, rectangles, pentagons), and we all know what polyhedra are (cubes, tetrahedrons, icosahedrons). Well a polytope is just a higher dimensional version of this, its just mathematicians got tired of affixing different words to “poly-” so they just say “polytopes.” You can make a polyhedron with an arbitrarily large face (for example a thousand sided polygon). It turns out if a polytope has dimension greater than 8, then it must have a “small” three dimensional face. One does not expect this to be the case, but it is. This theorem (Kalai admits) has absolutely no purpose. But it is one of those great little self contained mysteries of mathematics. 

The final image is the original text from Kalai’s paper.


theapocalypticalgorithm:

scienceisbeauty:
A Knot Zoo.

A formal “knot" can be thought of as a loop of string that is possibly knotted with itself. We can always draw such objects with the above diagrams in a way so that the knot lays flat with at most one segment lying above any other as seen from above. We can obviously rearrange the knot so that it lies with a different number of such crossings (put a kink in it for instance), but the least number of these crossings is called the crossing number. Mathematicians than ask themselves to classify the distinct knots (i.e. the knots that cannot be arranged to look identical). Above is a chart of all the distinct knots of crossing number up to nine. (the number above each knots indicates the crossing number, the subscripts are arbitrary counters)
The last few on the list are actually “links" which are multiple loops of rope intertwined. They too have crossing numbers. (again the number above each link indicates the crossing number, the subscript is a counter, and the superscript is the number of loops)

theapocalypticalgorithm:

scienceisbeauty:

A Knot Zoo.

A formal “knot" can be thought of as a loop of string that is possibly knotted with itself. We can always draw such objects with the above diagrams in a way so that the knot lays flat with at most one segment lying above any other as seen from above. We can obviously rearrange the knot so that it lies with a different number of such crossings (put a kink in it for instance), but the least number of these crossings is called the crossing number. Mathematicians than ask themselves to classify the distinct knots (i.e. the knots that cannot be arranged to look identical). Above is a chart of all the distinct knots of crossing number up to nine. (the number above each knots indicates the crossing number, the subscripts are arbitrary counters)

The last few on the list are actually “links" which are multiple loops of rope intertwined. They too have crossing numbers. (again the number above each link indicates the crossing number, the subscript is a counter, and the superscript is the number of loops)


a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. 

a11isun:

Allison Chen | Book of Tessellations | in progress

The dual of a tiling is obtained from placing a line perpendicular to each existing line so that all the lines perpendicular to the lines that bound a region meet at that regions center. Note that the dual of the dual is the original tiling. 


These doodles are taken from Milnor’s paper "Whitehead Torsion". In it, he sets out some basic algebraic constructions for chain complexes. He then goes on to prove that these constructions are a combinatorial invariant for cellular complexes (like the penguin from my previous post). It turns out this construct is essential in higher homotopy theory, though it can basically be defined as the determinant of a change of basis matrix. 
These types of pictures are very common for early topology papers, they were produced by hand. What’s so lovely about them is that they are representing arbitrary dimensions, and are somehow clear to those experts who would read this. Do not ask me what they mean, for, I have yet to make sense of the latter half of this paper. These doodles are taken from Milnor’s paper "Whitehead Torsion". In it, he sets out some basic algebraic constructions for chain complexes. He then goes on to prove that these constructions are a combinatorial invariant for cellular complexes (like the penguin from my previous post). It turns out this construct is essential in higher homotopy theory, though it can basically be defined as the determinant of a change of basis matrix. 
These types of pictures are very common for early topology papers, they were produced by hand. What’s so lovely about them is that they are representing arbitrary dimensions, and are somehow clear to those experts who would read this. Do not ask me what they mean, for, I have yet to make sense of the latter half of this paper. These doodles are taken from Milnor’s paper "Whitehead Torsion". In it, he sets out some basic algebraic constructions for chain complexes. He then goes on to prove that these constructions are a combinatorial invariant for cellular complexes (like the penguin from my previous post). It turns out this construct is essential in higher homotopy theory, though it can basically be defined as the determinant of a change of basis matrix. 
These types of pictures are very common for early topology papers, they were produced by hand. What’s so lovely about them is that they are representing arbitrary dimensions, and are somehow clear to those experts who would read this. Do not ask me what they mean, for, I have yet to make sense of the latter half of this paper. These doodles are taken from Milnor’s paper "Whitehead Torsion". In it, he sets out some basic algebraic constructions for chain complexes. He then goes on to prove that these constructions are a combinatorial invariant for cellular complexes (like the penguin from my previous post). It turns out this construct is essential in higher homotopy theory, though it can basically be defined as the determinant of a change of basis matrix. 
These types of pictures are very common for early topology papers, they were produced by hand. What’s so lovely about them is that they are representing arbitrary dimensions, and are somehow clear to those experts who would read this. Do not ask me what they mean, for, I have yet to make sense of the latter half of this paper. 

These doodles are taken from Milnor’s paper "Whitehead Torsion". In it, he sets out some basic algebraic constructions for chain complexes. He then goes on to prove that these constructions are a combinatorial invariant for cellular complexes (like the penguin from my previous post). It turns out this construct is essential in higher homotopy theory, though it can basically be defined as the determinant of a change of basis matrix. 

These types of pictures are very common for early topology papers, they were produced by hand. What’s so lovely about them is that they are representing arbitrary dimensions, and are somehow clear to those experts who would read this. Do not ask me what they mean, for, I have yet to make sense of the latter half of this paper.