Mathematical Structures From Linear Algebra over Rings to Geometry with Sheaves

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How to fold and cut a Christmas star<br /> Christian Lawson-Perfect https://www.youtube.com/watch?v=S90WPkgxvas

What a great simple example with some interesting complexity.

For teachers trying this with students, when one is done making some five pointed stars, the next questions a curious mathematician might ask are: how might I generalize this new knowledge to make a 6 pointed star? A 7 pointed star? a 1,729 pointed star? Is there a maximum number of points possible? Is there a minimum? Can any star be made without a cut? What happens if we make more than one cut? Are there certain numbers for which a star can't be made? Is there a relationship between the number of folds made and the number of points? What does all this have to do with our basic definition of what a paper star might look like? What other questions might we ask to extend this little idea of cutting paper stars?

Recalling some results from my third grade origami days, based on the thickness of most standard office paper, a typical sheet of paper can only be folded in half at most 7 times. This number can go up a bit if the thickness of the paper is reduced, but having a maximum number of potential folds suggests there is an upper bound for how many points a star might have using this method of construction.

Link to Ivan Sutherlands demo

The formulation uses concepts from the nascent field of discrete differential geometry

直到数学家 Bolyai、Lobachevsky 和 ​​Riemann 的出现。 Bolyai 试图推导出一个矛盾，假设通过一个点，平行于给定的线，可以画出许多线。他疯狂地推导却没有得到矛盾，最终意识到他发明了一种新的几何学，它不同于欧几里德的几何学，但同样值得尊敬。Riemann 走了另一条路。他假设通过一个点，平行于给定的线，不能画出任何线。他意识到他也发明了另一种几何学，实际上是球体上大圆的几何学。

One of the problems in approaching quantum gravity is the choice for how to best represent it mathematically. Most of quantum mechanics is algebraic in nature but gravity has a geometry component which is important. (restatement)

This is similar to the early 20th century problem of how to best represent quantum mechanics: as differential equations or using group theory/Lie algebras?

This prompts the question: what other potential representations might also work?

Could it be better understood/represented using Algebraic geometry or algebraic topology as perspectives?

[handwritten notes from 2023-02-02]

She doesn't mention it, but they're not only placing things in order for potential memory purposes, but they're also placing an order on their world as well.

Ladders and steps were frequently used to create an order of beings as in the scala naturae or the Great Chain of Being.

Some of this is also seen in Ramon Lull's Ladder of Ascent and Descent of the Mind, 1305 (Ars Magna)

Some of the effort that teachers put into their educational resources in an attempt to make learning faster and more efficient is actually taking away the actual learning opportunities of the students to sort, arrange, and make meaningful connections between the knowledge and to their own prior knowledge bases.

In mathematics, rather than showing a handful of methods for solving a problem, the teacher might help students to explore those problem solving spaces first and then assist them into creating these algorithms. I can't help but think about Inventional Geometry by William George Spencer that is structured this way. The teacher has created a broader super-structure of problems, but leaves it largely to the student to do the majority of the work.

https://www.scientificamerican.com/article/an-ancient-greek-astronomical-calculation-machine-reveals-new-secrets/

Overview and history of the Antikythera mechanism and the current state of research surrounding it.

Antikythera mechanism found in diving expedition in 1900 by Elias Stadiatis. It was later dated between 60 and 70 BCE, but evidence suggests it may have been made around 205 BCE.

Functions

One of the primary purposes of the device was to predict the positions of the planets along the ecliptic, the plane of the solar system.

The device was also used to track the positions of the sun and moon. This included the moon's phase, position and age (the number of days from a new moon). It also included the predictions of eclipses.

Used to track the motions of the 5 known planets including 289 synodic cycles in 462 years for Venus and 427 synodic cycles in 442 years for Saturn.

Risings and settings of stars indexed to a zodiac dial

Definitions

metonic cycle, a 19-year period over which 235 moon phases recur; named after Greek astronomer Meton, but discovered much earlier by the Babylonians. The Greeks refined it to a 76 year period.

saros cycle, the 223 month lunar cycle which was used by the Babylonians to predict eclipses. A dial on the Antikythera mechanism was used to predict the dates of the solar and lunar eclipses using this cycle.

synodic events: conjunctions with the sun and its stationary points

People

Archimedes - potentially the designer of an early version of the Antikythera mechanism

Elias Stadiatis - diver who discovered the Antikythera mechanism

Albert Rehm - German philologist who the numbers 19, 76 and 223 inscribed on fragments of the device in the early 1900s

Derek J. de Solla Price, published Gears from the Greeks in 1974. Identified the gear train and developed a complete model of the gearing.

Michael Wright - 3D x-ray study in 1990 using linear tomography; identified tooth counts of the gears and understood the upper dial on the back of the device

Tony Freeth - author of article and researcher whose made recent discoveries.

https://output.jsbin.com/rifoyu

One of the better "simple" discussions of dual spaces I've seen:

If you have a vector space, any vector space, you can define linear functions on that space. The set of all those functions is the dual space of the vector space. The important point here is that it doesn't matter what this original vector space is. You have a vector space V, you have a corresponding dual V∗.

OK, now you have linear functions. Now if you add two linear functions, you get again a linear function. Also if you multiply a linear function with a factor, you get again a linear function. Indeed, you can check that linear functions fulfill all the vector space axioms this way. Or in short, the dual space is a vector space in its own right.

But if V∗ is a vector space, then it comes with everything a vector space comes with. But as we have seen in the beginning, one thing every vector space comes with is a dual space, the space of all linear functions on it. Therefore also the dual space V∗ has a corresponding dual space, V∗∗, which is called double dual space (because "dual space of the dual space" is a bit long).

So we have the dual space, but we also want to know what sort of functions are in that double dual space. Well, such a function takes a vector from V∗, that is, a linear function on V, and maps that to a scalar (that is, to a member of the field the vector space is based on). Now, if you have a linear function on V, you already know a way to get a scalar from that: Just apply it to a vector from V. Indeed, it is not hard to show that if you just choose an arbitrary fixed element v∈V, then the function Fv:ϕ↦ϕ(v) indeed is a linear function on V∗, and thus a member of the double dual V∗∗. That way we have not only identified certain members of V∗∗ but in addition a natural mapping from V to V∗∗, namely F:v↦Fv. It is not hard to prove that this mapping is linear and injective, so that the functions in V∗∗ corresponding to vectors in V form a subspace of V∗∗. Indeed, if V is finite dimensional, it's even all of V∗∗. That's easy to see if you know that dim(V∗)=dimV and therefore dim(V∗∗)=dimV∗=dimV. On the other hand, since F is injective, dim(F(V))=dim(V). However for finite dimensional vector spaces, the only subspace of the same dimension as the full space is the full space itself. However if V is infinite dimensional, V∗∗ is larger than V. In other words, there are functions in V∗∗ which are not of the form Fv with v∈V.

Note that since V∗∗again is a vector space, it also has a dual space, which again has a dual space, and so on. So in principle you have an infinite series of duals (although only for infinite vector spaces they are all different).

Note to self: Look at [[Steven Roman]]'s book [[The Umbral Calculus]] to follow up on having a more intuitive idea of what a dual space is and how it's useful

Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R where M is a smooth manifold is an object that produces, for any point p∈M and tangent vector v∈TpM, a number, the directional derivative, in a linear way. In other words, ==a differentiable function defines an element of the dual to the tangent space (the cotangent space) at each point of the manifold.==

A more intuitive description of why dual spaces are useful or interesting.

Differential topology take a more global view and studies structures on manifolds that have no interesting local structure while differential geometry studies structures on manifolds that have interesting local structures.

Joyce seems to like putting elements from mathematics in his works. He referred the "Ithaca" episode of Ulysses as a "mathematical catechism" in his letter. Maybe he found some connections between geometry and Catechism as he was a very knowledgeable writer.

Qian, Y., Expert, P., Panzarasa, P., & Barahona, M. (2020). Geometric graphs from data to aid classification tasks with graph convolutional networks. ArXiv:2005.04081 [Physics, Stat]. http://arxiv.org/abs/2005.04081

Calculating the Area of a Triangle

Decomposing rectilinear shapes to find an area.

More here.

就喜欢这些用微分流形讲机器学习的~！

应该写个 Paper Summary 表示尊敬~~~

一篇Physical Review X 上的文章~ 读读看~

Paper Summary

The Use of Projective Geometry in Computer Graphics

Chapter 2: 射影几何的一般性介绍

Para o nosso caso, assumir essa hipótese com respeito a decomposição do espaço tangente ao longo do bordo.

Qual a relação dessa desigualdade com a dita desigualdade de Penrose Riemanniana provada por Huisken-Ilmanen e Bray?

Qual a relação dessa desigualdade com a dita desigualdade de Penrose Riemanniana provada por Huisken-Ilmanen e Bray?

comparison between geometry and rhetoric