nhaliday + geometry   123

John Dee - Wikipedia
John Dee (13 July 1527 – 1608 or 1609) was an English mathematician, astronomer, astrologer, occult philosopher,[5] and advisor to Queen Elizabeth I. He devoted much of his life to the study of alchemy, divination, and Hermetic philosophy. He was also an advocate of England's imperial expansion into a "British Empire", a term he is generally credited with coining.[6]

Dee straddled the worlds of modern science and magic just as the former was emerging. One of the most learned men of his age, he had been invited to lecture on the geometry of Euclid at the University of Paris while still in his early twenties. Dee was an ardent promoter of mathematics and a respected astronomer, as well as a leading expert in navigation, having trained many of those who would conduct England's voyages of discovery.

Simultaneously with these efforts, Dee immersed himself in the worlds of magic, astrology and Hermetic philosophy. He devoted much time and effort in the last thirty years or so of his life to attempting to commune with angels in order to learn the universal language of creation and bring about the pre-apocalyptic unity of mankind. However, Robert Hooke suggested in the chapter Of Dr. Dee's Book of Spirits, that John Dee made use of Trithemian steganography, to conceal his communication with Elizabeth I.[7] A student of the Renaissance Neo-Platonism of Marsilio Ficino, Dee did not draw distinctions between his mathematical research and his investigations into Hermetic magic, angel summoning and divination. Instead he considered all of his activities to constitute different facets of the same quest: the search for a transcendent understanding of the divine forms which underlie the visible world, which Dee called "pure verities".

In his lifetime, Dee amassed one of the largest libraries in England. His high status as a scholar also allowed him to play a role in Elizabethan politics. He served as an occasional advisor and tutor to Elizabeth I and nurtured relationships with her ministers Francis Walsingham and William Cecil. Dee also tutored and enjoyed patronage relationships with Sir Philip Sidney, his uncle Robert Dudley, 1st Earl of Leicester, and Edward Dyer. He also enjoyed patronage from Sir Christopher Hatton.

https://twitter.com/Logo_Daedalus/status/985203144044040192
https://archive.is/h7ibQ
mind meld

Leave Me Alone! Misanthropic Writings from the Anti-Social Edge
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april 2018 by nhaliday
Ultimate fate of the universe - Wikipedia
The fate of the universe is determined by its density. The preponderance of evidence to date, based on measurements of the rate of expansion and the mass density, favors a universe that will continue to expand indefinitely, resulting in the "Big Freeze" scenario below.[8] However, observations are not conclusive, and alternative models are still possible.[9]

Big Freeze or heat death
Main articles: Future of an expanding universe and Heat death of the universe
The Big Freeze is a scenario under which continued expansion results in a universe that asymptotically approaches absolute zero temperature.[10] This scenario, in combination with the Big Rip scenario, is currently gaining ground as the most important hypothesis.[11] It could, in the absence of dark energy, occur only under a flat or hyperbolic geometry. With a positive cosmological constant, it could also occur in a closed universe. In this scenario, stars are expected to form normally for 1012 to 1014 (1–100 trillion) years, but eventually the supply of gas needed for star formation will be exhausted. As existing stars run out of fuel and cease to shine, the universe will slowly and inexorably grow darker. Eventually black holes will dominate the universe, which themselves will disappear over time as they emit Hawking radiation.[12] Over infinite time, there would be a spontaneous entropy decrease by the Poincaré recurrence theorem, thermal fluctuations,[13][14] and the fluctuation theorem.[15][16]

A related scenario is heat death, which states that the universe goes to a state of maximum entropy in which everything is evenly distributed and there are no gradients—which are needed to sustain information processing, one form of which is life. The heat death scenario is compatible with any of the three spatial models, but requires that the universe reach an eventual temperature minimum.[17]
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april 2018 by nhaliday
Antinomia Imediata – experiments in a reaction from the left
https://antinomiaimediata.wordpress.com/lrx/
So, what is the Left Reaction? First of all, it’s reaction: opposition to the modern rationalist establishment, the Cathedral. It opposes the universalist Jacobin program of global government, favoring a fractured geopolitics organized through long-evolved complex systems. It’s profoundly anti-socialist and anti-communist, favoring market economy and individualism. It abhors tribalism and seeks a realistic plan for dismantling it (primarily informed by HBD and HBE). It looks at modernity as a degenerative ratchet, whose only way out is intensification (hence clinging to crypto-marxist market-driven acceleration).

How come can any of this still be in the *Left*? It defends equality of power, i.e. freedom. This radical understanding of liberty is deeply rooted in leftist tradition and has been consistently abhored by the Right. LRx is not democrat, is not socialist, is not progressist and is not even liberal (in its current, American use). But it defends equality of power. It’s utopia is individual sovereignty. It’s method is paleo-agorism. The anti-hierarchy of hunter-gatherer nomads is its understanding of the only realistic objective of equality.

...

In more cosmic terms, it seeks only to fulfill the Revolution’s side in the left-right intelligence pump: mutation or creation of paths. Proudhon’s antinomy is essentially about this: the collective force of the socius, evinced in moral standards and social organization vs the creative force of the individuals, that constantly revolutionize and disrupt the social body. The interplay of these forces create reality (it’s a metaphysics indeed): the Absolute (socius) builds so that the (individualistic) Revolution can destroy so that the Absolute may adapt, and then repeat. The good old formula of ‘solve et coagula’.

Ultimately, if the Neoreaction promises eternal hell, the LRx sneers “but Satan is with us”.

https://antinomiaimediata.wordpress.com/2016/12/16/a-statement-of-principles/
Liberty is to be understood as the ability and right of all sentient beings to dispose of their persons and the fruits of their labor, and nothing else, as they see fit. This stems from their self-awareness and their ability to control and choose the content of their actions.

...

Equality is to be understood as the state of no imbalance of power, that is, of no subjection to another sentient being. This stems from their universal ability for empathy, and from their equal ability for reason.

...

It is important to notice that, contrary to usual statements of these two principles, my standpoint is that Liberty and Equality here are not merely compatible, meaning they could coexist in some possible universe, but rather they are two sides of the same coin, complementary and interdependent. There can be NO Liberty where there is no Equality, for the imbalance of power, the state of subjection, will render sentient beings unable to dispose of their persons and the fruits of their labor[1], and it will limit their ability to choose over their rightful jurisdiction. Likewise, there can be NO Equality without Liberty, for restraining sentient beings’ ability to choose and dispose of their persons and fruits of labor will render some more powerful than the rest, and establish a state of subjection.

https://antinomiaimediata.wordpress.com/2017/04/18/flatness/
equality is the founding principle (and ultimately indistinguishable from) freedom. of course, it’s only in one specific sense of “equality” that this sentence is true.

to try and eliminate the bullshit, let’s turn to networks again:

any nodes’ degrees of freedom is the number of nodes they are connected to in a network. freedom is maximum when the network is symmetrically connected, i. e., when all nodes are connected to each other and thus there is no topographical hierarchy (middlemen) – in other words, flatness.

in this understanding, the maximization of freedom is the maximization of entropy production, that is, of intelligence. As Land puts it:

https://antinomiaimediata.wordpress.com/category/philosophy/mutualism/
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march 2018 by nhaliday
Hyperbolic angle - Wikipedia
A unit circle {\displaystyle x^{2}+y^{2}=1} x^2 + y^2 = 1 has a circular sector with an area half of the circular angle in radians. Analogously, a unit hyperbola {\displaystyle x^{2}-y^{2}=1} {\displaystyle x^{2}-y^{2}=1} has a hyperbolic sector with an area half of the hyperbolic angle.
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november 2017 by nhaliday
What is the connection between special and general relativity? - Physics Stack Exchange
Special relativity is the "special case" of general relativity where spacetime is flat. The speed of light is essential to both.
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november 2017 by nhaliday
What is the difference between general and special relativity? - Quora
General Relativity is, quite simply, needed to explain gravity.

Special Relativity is the special case of GR, when the metric is flat — which means no gravity.

You need General Relativity when the metric gets all curvy, and when things start to experience gravitation.
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november 2017 by nhaliday
gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
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october 2017 by nhaliday
Best Topology Olympiad ***EVER*** - Affine Mess - Quora
Most people take courses in topology, algebraic topology, knot theory, differential topology and what have you without once doing anything with a finite topological space. There may have been some quirky questions about such spaces early on in a point-set topology course, but most of us come out of these courses thinking that finite topological spaces are either discrete or only useful as an exotic counterexample to some standard separation property. The mere idea of calculating the fundamental group for a 4-point space seems ludicrous.

Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!
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october 2017 by nhaliday
Power of a point - Wikipedia
The power of point P (see in Figure 1) can be defined equivalently as the product of distances from the point P to the two intersection points of any ray emanating from P.
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september 2017 by nhaliday
newtonian gravity - Newton's original proof of gravitation for non-point-mass objects - Physics Stack Exchange
This theorem is Proposition LXXI, Theorem XXXI in the Principia. To warm up, consider the more straightforward proof of the preceding theorem, that there's no inverse-square force inside of a spherical shell:

picture

The crux of the argument is that the triangles HPI and LPK are similar. The mass enclosed in the small-but-near patch of sphere HI goes like the square of the distance HP, while the mass enclosed in the large-but-far patch of sphere KL, with the same solid angle, goes like the square of the distance KP. This mass ratio cancels out the distance-squared ratio governing the strength of the force, and so the net force from those two patches vanishes.

For a point mass outside a shell, Newton's approach is essentially the same as the modern approach:

picture

One integral is removed because we're considering a thin spherical shell rather than a solid sphere. The second integral, "as the semi-circle AKB revolves about the diameter AB," trivially turns Newton's infinitesimal arcs HI and KL into annuli.

The third integral is over all the annuli in the sphere, over 0≤ϕ≤τ/20≤ϕ≤τ/2 or over R−r≤s≤R+rR−r≤s≤R+r. This one is a little bit hairy, even with the advantage of modern notation.

Newton's clever trick is to consider the relationship between the force due to the smaller, nearer annulus HI and the larger, farther annulus KL defined by the same viewing angle (in modern notation, dθdθ). If I understand correctly he argues again, based on lots of similar triangles with infinitesimal angles, that the smaller-but-nearer annulus and the larger-but-farther annulus exert the same force at P. Furthermore, he shows that the force doesn't depend on the distance PF, and thus doesn't depend on the radius of the sphere; the only parameter left is the distance PS (squared) between the particle and the sphere's center. Since the argument doesn't depend on the angle HPS, it's true for all the annuli, and the theorem is proved.
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september 2017 by nhaliday
Inscribed angle - Wikipedia
pf:
- for triangle w/ one side = a diameter, draw isosceles triangle and use supplementary angle identities
- otherwise draw second triangle w/ side = a diameter, and use above result twice
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august 2017 by nhaliday
co.combinatorics - Classification of Platonic solids - MathOverflow
My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula v−e+f=2v−e+f=2 to show that there are exactly five possible triples (v,e,f)(v,e,f). But of course this is not a complete proof because it does not rule out the possibility of different configurations or deformations. Has anyone ever written up a complete proof of this statement?!

...

This is a classical question. Here is my reading of it: Why is there a unique polytope with given combinatorics of faces, which are all regular polygons? Of course, for simple polytopes (tetrahedron, cube, dodecahedron) this is clear, but for the octahedron and icosahedron this is less clear.

The answer lies in the Cauchy's theorem. It was Legendre, while writing his Elements of Geometry and Trigonometry, noticed that Euclid's proof is incomplete in the Elements. Curiously, Euclid finds both radii of inscribed and circumscribed spheres (correctly) without ever explaining why they exist. Cauchy worked out a proof while still a student in 1813, more or less specifically for this purpose. The proof also had a technical gap which was found and patched up by Steinitz in 1920s.

The complete (corrected) proof can be found in the celebrated Proofs from the Book, or in Marcel Berger's Geometry. My book gives a bit more of historical context and some soft arguments (ch. 19). It's worth comparing this proof with (an erroneous) pre-Steinitz exposition, say in Hadamard's Leçons de Géométrie Elémentaire II, or with an early post-Steinitz correct but tedious proof given in (otherwise, excellent) Alexandrov's monograph (see also ch.26 in my book which compares all the approaches).

P.S. Note that Coxeter in Regular Polytopes can completely avoid this issue but taking a different (modern) definition of the regular polytopes (which are symmetric under group actions). For a modern exposition and the state of art of this approach, see McMullen and Schulte's Abstract Regular Polytopes.

https://en.wikipedia.org/wiki/Platonic_solid#Classification
https://mathoverflow.net/questions/46502/on-the-number-of-archimedean-solids
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july 2017 by nhaliday
If there are 3 space dimensions and one time dimension, is it theoretically possible to have multiple time demensions and if so how would it work? : askscience
Yes, we can consider spacetimes with any number of temporal or spatial dimensions. The theory is set up essentially the same. Spacetime is modeled as a smooth n-dimensional manifold with a pseudo-Riemannian metric, and the metric satisfies the Einstein field equations (Einstein tensor = stress tensor).
A pseudo-Riemannian tensor is characterized by its signature, i.e., the number of negative quadratic forms in its metric and the number of positive quadratic forms. The coordinates with negative forms correspond to temporal dimensions. (This is a convention that is fixed from the start.) In general relativity, spacetime is 4-dimensional, and the signature is (1,3), so there is 1 temporal dimension and 3 spatial dimensions.
Okay, so that's a lot of math, but it all basically means that, yes, it makes sense to ask questions like "what does a universe with 2 time dimensions and 3 spatial dimensions look like?" It turns out that spacetimes with more than 1 temporal dimension are very pathological. For one, initial value problems do not generally have unique solutions. There is also generally no canonical way to pick out 1 of the infinitely many solutions to the equations of physics. This means that predictability is impossible (e.g., how do you know which solution is the correct one?). Essentially, there is no meaningful physics in a spacetime with more than 1 temporal dimension.
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june 2017 by nhaliday
Lucio Russo - Wikipedia
In The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (Italian: La rivoluzione dimenticata), Russo promotes the belief that Hellenistic science in the period 320-144 BC reached heights not achieved by Classical age science, and proposes that it went further than ordinarily thought, in multiple fields not normally associated with ancient science.

La Rivoluzione Dimenticata (The Forgotten Revolution), Reviewed by Sandro Graffi: http://www.ams.org/notices/199805/review-graffi.pdf

Before turning to the question of the decline of Hellenistic science, I come back to the new light shed by the book on Euclid’s Elements and on pre-Ptolemaic astronomy. Euclid’s definitions of the elementary geometric entities—point, straight line, plane—at the beginning of the Elements have long presented a problem.7 Their nature is in sharp contrast with the approach taken in the rest of the book, and continued by mathematicians ever since, of refraining from defining the fundamental entities explicitly but limiting themselves to postulating the properties which they enjoy. Why should Euclid be so hopelessly obscure right at the beginning and so smooth just after? The answer is: the definitions are not Euclid’s. Toward the beginning of the second century A.D. Heron of Alexandria found it convenient to introduce definitions of the elementary objects (a sign of decadence!) in his commentary on Euclid’s Elements, which had been written at least 400 years before. All manuscripts of the Elements copied ever since included Heron’s definitions without mention, whence their attribution to Euclid himself. The philological evidence leading to this conclusion is quite convincing.8

...

What about the general and steady (on the average) impoverishment of Hellenistic science under the Roman empire? This is a major historical problem, strongly tied to the even bigger one of the decline and fall of the antique civilization itself. I would summarize the author’s argument by saying that it basically represents an application to science of a widely accepted general theory on decadence of antique civilization going back to Max Weber. Roman society, mainly based on slave labor, underwent an ultimately unrecoverable crisis as the traditional sources of that labor force, essentially wars, progressively dried up. To save basic farming, the remaining slaves were promoted to be serfs, and poor free peasants reduced to serfdom, but this made trade disappear. A society in which production is almost entirely based on serfdom and with no trade clearly has very little need of culture, including science and technology. As Max Weber pointed out, when trade vanished, so did the marble splendor of the ancient towns, as well as the spiritual assets that went with it: art, literature, science, and sophisticated commercial laws. The recovery of Hellenistic science then had to wait until the disappearance of serfdom at the end of the Middle Ages. To quote Max Weber: “Only then with renewed vigor did the old giant rise up again.”

...

The epilogue contains the (rather pessimistic) views of the author on the future of science, threatened by the apparent triumph of today’s vogue of irrationality even in leading institutions (e.g., an astrology professorship at the Sorbonne). He looks at today’s ever-increasing tendency to teach science more on a fideistic than on a deductive or experimental basis as the first sign of a decline which could be analogous to the post-Hellenistic one.

Praising Alexandrians to excess: https://sci-hub.tw/10.1088/2058-7058/17/4/35
The Economic Record review: https://sci-hub.tw/10.1111/j.1475-4932.2004.00203.x

listed here: https://pinboard.in/u:nhaliday/b:c5c09f2687c1

Was Roman Science in Decline? (Excerpt from My New Book): https://www.richardcarrier.info/archives/13477
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may 2017 by nhaliday
how big was the edge? | West Hunter
random side note:
- dysgenics running at -.5-1 IQ/generation in NW Europe since ~1800 and China by ~1960
- gap between east asians and europeans typically a bit less than .5 SD (or .3 SD if you look at mainland chinese not asian-americans?), similar variances
- 160/30 * 1/15 = .36, so could explain most of gap depending on when exactly dysgenics started
- maybe Europeans were just smarter back then? still seems like you need additional cultural/personality and historical factors. could be parasite load too.
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march 2017 by nhaliday
A Unified Theory of Randomness | Quanta Magazine
Beyond the one-dimensional random walk, there are many other kinds of random shapes. There are varieties of random paths, random two-dimensional surfaces, random growth models that approximate, for example, the way a lichen spreads on a rock. All of these shapes emerge naturally in the physical world, yet until recently they’ve existed beyond the boundaries of rigorous mathematical thought. Given a large collection of random paths or random two-dimensional shapes, mathematicians would have been at a loss to say much about what these random objects shared in common.

Yet in work over the past few years, Sheffield and his frequent collaborator, Jason Miller, a professor at the University of Cambridge, have shown that these random shapes can be categorized into various classes, that these classes have distinct properties of their own, and that some kinds of random objects have surprisingly clear connections with other kinds of random objects. Their work forms the beginning of a unified theory of geometric randomness.
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february 2017 by nhaliday
Riemannian manifold - Wikipedia
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product {\displaystyle g_{p}} on the tangent space {\displaystyle T_{p}M} at each point {\displaystyle p} that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then {\displaystyle p\mapsto g_{p}(X(p),Y(p))} is a smooth function. The family {\displaystyle g_{p}} of inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

A Riemannian metric (tensor) makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields.
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february 2017 by nhaliday
Orthogonal — Greg Egan
In Yalda’s universe, light has no universal speed and its creation generates energy.

On Yalda’s world, plants make food by emitting their own light into the dark night sky.
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february 2017 by nhaliday
Prékopa–Leindler inequality | Academically Interesting
Consider the following statements:
1. The shape with the largest volume enclosed by a given surface area is the n-dimensional sphere.
2. A marginal or sum of log-concave distributions is log-concave.
3. Any Lipschitz function of a standard n-dimensional Gaussian distribution concentrates around its mean.
What do these all have in common? Despite being fairly non-trivial and deep results, they all can be proved in less than half of a page using the Prékopa–Leindler inequality.

ie, Brunn-Minkowski
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february 2017 by nhaliday
The Brunn-Minkowski Inequality | The n-Category Café
For instance, this happens in the plane when A is a horizontal line segment and B is a vertical line segment. There’s obviously no hope of getting an equation for Vol(A+B) in terms of Vol(A) and Vol(B). But this example suggests that we might be able to get an inequality, stating that Vol(A+B) is at least as big as some function of Vol(A) and Vol(B).

The Brunn-Minkowski inequality does this, but it’s really about linearized volume, Vol^{1/n}, rather than volume itself. If length is measured in metres then so is Vol^{1/n}.

...

Nice post, Tom. To readers whose background isn’t in certain areas of geometry and analysis, it’s not obvious that the Brunn–Minkowski inequality is more than a curiosity, the proof of the isoperimetric inequality notwithstanding. So let me add that Brunn–Minkowski is an absolutely vital tool in many parts of geometry, analysis, and probability theory, with extremely diverse applications. Gardner’s survey is a great place to start, but by no means exhaustive.

I’ll also add a couple remarks about regularity issues. You point out that Brunn–Minkowski holds “in the vast generality of measurable sets”, but it may not be initially obvious that this needs to be interpreted as “when A, B, and A+B are all Lebesgue measurable”, since A+B need not be measurable when A and B are (although you can modify the definition of A+B to work for arbitrary measurable A and B; this is discussed by Gardner).
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february 2017 by nhaliday
mg.metric geometry - Pushing convex bodies together - MathOverflow
- volume of intersection of colliding, constant-velocity convex bodies is unimodal
- pf by Brunn-Minkowski inequality
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january 2017 by nhaliday
Ehrhart polynomial - Wikipedia
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.
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january 2017 by nhaliday
Mikhail Leonidovich Gromov - Wikipedia
Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.

Gromov is also interested in mathematical biology,[11] the structure of the brain and the thinking process, and the way scientific ideas evolve.[8]
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january 2017 by nhaliday
Dvoretzky's theorem - Wikipedia
In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids.

http://mathoverflow.net/questions/143527/intuitive-explanation-of-dvoretzkys-theorem
http://mathoverflow.net/questions/46278/unexpected-applications-of-dvoretzkys-theorem
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january 2017 by nhaliday
Carathéodory's theorem (convex hull) - Wikipedia
- any convex combination in R^d can be pared down to at most d+1 points
- eg, in R^2 you can always fit a point in convex hull in a triangle
tcs  acm  math.MG  geometry  levers  wiki  reference  optimization  linear-programming  math  linear-algebra  nibble  spatial  curvature  convexity-curvature 
january 2017 by nhaliday
(Gil Kalai) The weak epsilon-net problem | What's new
This is a problem in discrete and convex geometry. It seeks to quantify the intuitively obvious fact that large convex bodies are so “fat” that they cannot avoid “detection” by a small number of observation points.
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january 2017 by nhaliday
ho.history overview - Proofs that require fundamentally new ways of thinking - MathOverflow
my favorite:
Although this has already been said elsewhere on MathOverflow, I think it's worth repeating that Gromov is someone who has arguably introduced more radical thoughts into mathematics than anyone else. Examples involving groups with polynomial growth and holomorphic curves have already been cited in other answers to this question. I have two other obvious ones but there are many more.

I don't remember where I first learned about convergence of Riemannian manifolds, but I had to laugh because there's no way I would have ever conceived of a notion. To be fair, all of the groundwork for this was laid out in Cheeger's thesis, but it was Gromov who reformulated everything as a convergence theorem and recognized its power.

Another time Gromov made me laugh was when I was reading what little I could understand of his book Partial Differential Relations. This book is probably full of radical ideas that I don't understand. The one I did was his approach to solving the linearized isometric embedding equation. His radical, absurd, but elementary idea was that if the system is sufficiently underdetermined, then the linear partial differential operator could be inverted by another linear partial differential operator. Both the statement and proof are for me the funniest in mathematics. Most of us view solving PDE's as something that requires hard work, involving analysis and estimates, and Gromov manages to do it using only elementary linear algebra. This then allows him to establish the existence of isometric embedding of Riemannian manifolds in a wide variety of settings.
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january 2017 by nhaliday
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