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For those of you interested in modern and contemporary art, Art Resource recently added around 750 new images to JSTOR–featuring iconic figures like Georgia O'Keefe and Jacob Lawrence.

Learn more about the collection in our recent blog post.

Image: Wayne Thiebaud. Pie Counter, 1963. Oil on canvas. © Whitney Museum of American Art / Licensed by Scala / Art Resource, NY

[mathhombre]

Aspirational Numbers

Love number theory like this. Accessible, new to me, perfect. Also abundant and deficient.

AND Ben shared his GeoGebra applet.

[mathhombre]

@garadinervi sent it! Thank you!

Not what I was expecting.

Enzo Mari, Studio per il programma di Struttura 247, 1956 [Galleria Massimo Minini, Brescia. © Enzo Mari]

[mathhombre]

Front: 5 platonic solids and the 13 Archimedean solids

Back: Root-like structures, depicting some relationships between these 18 solids.

Both simultaneously (Holding it against light source):

[mathhombre]

@garadinervi also shared Struttura 245, but I can't find 247...

Makes you curious!

Enzo Mari, Studio per il programma di Struttura 247, 1956 [Galleria Massimo Minini, Brescia. © Enzo Mari]

[mathhombre]

Cat and Bird

Sweet positive/negative space tiling from Anna Zajac, posted in the FB Mathematical Tiling and Tessellations group.

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[mathhombre]

For example, 7x3=21, 5x9=45, etc. If you do, what is it?

–

We ask your questions so you don’t have to! Submit your questions to have them posted anonymously as polls.

[mathhombre]

Roman Dodecahedron

Richard Parker, secretary of the Norton Disney History and Archaeology Group, said it was a "privilege to have handled" the object, thought to have been buried about 1,700 years ago. However, he said: "Despite all the research that has gone into our dodecahedron, and others like it, we are no closer to finding out exactly what it is and what it might have been used for. "The imagination races when thinking about what the Romans may have used it for. Magic, rituals or religion - we perhaps may never know. "What we do know is the Norton Disney dodecahedron was found on the top of a hill in a former large pit of some kind. It seems it was deliberately placed there."

Fascinating story.

[mathhombre]

Ghee Beom Kim

Ghee is doing some cool circle tilings in the FB Mathematical Tiling and Tessellation group.

GBK has a fun IG as well.

[mathhombre]

[mathhombre]

The Swan, No. 1 by Hilma af Klint is one of her most well known works.

[mathhombre]

Victor Vasarely (1908-1997) [France] - ‘Pint Opus 0642’, 1965. Acrylic on wood (32 × 32 cm).

[mathhombre]

Victor Vasarely (1908-1997) [France] - ‘Noir-Vert-Bleu’, 1964-67. Tempera on wood, mounted on wooden panel (75 × 75 x 5 cm).

[mathhombre]

Line Bundles on Complex Tori

That's the hexagonal tiling honeycomb by Roice Nelson. John Carlos Baez is working through a series on these beautiful infinite spaces.

Order 5 cubic honeycomb.

Coxeter showed there are 15 regular hyperbolic honeycombs.

Fun article in a good series. Highly recommend if it's at all of interest.

[mathhombre]

Dadamaino Lines

Also tried a sine variation...

Which I like.

Not entirely what I'm looking for, but a good first swing to me. On GeoGebra if you want to play but the sine interpolations aren't working there.

Messed around with Dadamaino before, should look up more of their work.

Original via @garadinervi

[mathhombre]

Topological Spaces 12: Important Examples

We have spent quite a while developing a lot of theory and now it is time to apply it to some very important spaces that show up time and time again in topology!

Here is the link to the previous post about quotient spaces. They show up a lot in what follows, as does a lot of our theory so I highly recommend being familiar with the previous posts before reading this one!

12.1: The n-Sphere, Sⁿ

Definition 12.1:

A 2-sphere:

Proposition 12.2:

Remark: The last part of the proof fails for n=0 since ℝ\{0} is not connected. And indeed, S⁰={-1,1} is not connected (in this topology).

12.2: The n-Torus, Tⁿ

Defintion 12.3:

A (2-)torus:

Proposition 12.4:

12.3: Identifaction Squares

A particularly useful diagram in topology is the identification diagram which shows [0,1]×[0,1] as a square with arrows to indicate how to glue sections of the boundary together. The number of arrows indicated which sides to glue together (no arrows means that section is not glued to anything) and the direction corresponds to which direction the sections should be glued together. This gluing can we formalised with the use of an equivalence relation.

Since [0,1]×[0,1] is compact and path connected, our resulting spaces will also be compact and path connected.

Example 12.5:

As the picture suggests, M is homeomorphic to a subset of ℝ³ meaning it inherets Hausdorffness and second countability. For sake of brievity, the proof is omitted.

Example 12.6:

Example 12.7:

It is possible to show that K is homeomorphic to a subspace of ℝ⁴ so it is Hausdorff and second countable. It is also possible to show that it cannot be homeomorphic to a subspace of ℝ³ but this requires more advanced theory.

Example 12.8:

Just like with K, it is possible to show that P is homeomorphic ℝ⁴ so it is Hausdorff and second countable, but is not homeomorphic to a subspace of ℝ³.

12.4: Real Projective Space

Definition 12.9:

ℝℙⁿ is path connected and compact by Prop 11.5 and Prop 12.2. It is possbile to show ℝℙⁿ is Hausdorff and second countable but this takes a little more thoery.

It is also possible to show that P≅ℝℙ².

The next and final post will be a summary of what we've seen as well as hints at what else there is to topology!