euclidean space

While not all graphs can be drawn in R2, every single finite graph can be drawn in the 3 dimensional space R3. The example I will use is called a book embedding.

Imagine you put all of the vertices on the same line in R3. There are an infinite number of planes that go through every point on that line, and do not overlap anywhere else.

You can put each edge on a distinct plane, and they do not overlap, so it is a valid embedding in R3.

In fact, you don’t need to have one plane for each edge. You can put multiple edges on the same plane and they still don’t cross each other.

The minimum number of pages you need to embed a graph is constant no matter which order you put the vertices on the line.

“The image is indivisible and elusive, dependent upon our consciousness and on the real world which it seeks to embody. If the world is inscrutable, then the image will be so too. It is a kind of equation, signifying the correlation between truth and the human consciousness, bound as the latter is by Euclidean space. We cannot comprehend the totality of the universe, but the poetic image is able to express that totality.”

Andrei Tarkovsky, from “The film image,” Sculpting in Time, trans. Kitty Hunter-Blair (University of Texas Press, 1987)

mynamesvortex  asked:

THE REST OF THE TOWER NOW KNOWS THE TRAVELLER TASTES LIKE VANILLA. THERE IS NO WAY OUT NOW.

Wha… what? I said that it SMELLS faintly of vanilla. You should NEVER lick the Traveller. Non-Euclidean geometry of space (and all that) can’t be good for your tastebuds.

anonymous asked:

Could you explain this tfw no ZF joke? I really dont get it... :D

Get ready for a long explanation! For everyone’s reference, the joke (supplied by @awesomepus​) was:

Q: What did the mathematician say when he encountered the paradoxes of naive set theory?
A: tfw no ZF

You probably already know the ‘tfw no gf’ (that feel when no girlfriend) meme, which dates to 2010. I’m assuming you’re asking about the ZF part.

Mathematically, ZF is a reference to Zermelo-Fraenkel set theory, which is a set of axioms commonly accepted by mathematicians as the foundation of modern mathematics. As you probably know if you’ve taken geometry, axioms are super important: they are basic assumptions we make about the world we’re working in, and they have serious implications for what we can and can’t do in that world. 

For example, if you don’t assume the Parallel Postulate (that consecutive interior angle measures between two parallel lines and a transversal sum to 180°, or twice the size of a right angle), you can’t prove the Triangle Angle Sum Theorem (that the sum of the angle measures in any triangle is also 180°). It’s not that the Triangle Angle Sum Theorem theorem is not true without the Parallel Postulate — simply that it is unprovable, or put differently, neither true nor false, without that Postulate. Asking whether the Triangle Angle Sum Theorem is true without the Parallel Postulate is really a meaningless question, mathematically. But we understand that, in Euclidean geometry (not in curved geometries), both the postulate and the theorem are “true” in the sense that we have good reason to believe them (e.g., measuring lots of angles in physical parallel lines and triangles). Clearly, the axioms we choose are important.

Now, in the late 19th and early 20th century, mathematicians and logicians were interested in understanding the underpinnings of the basic structures we use in math — sets, or “collections,” being one of them, and arithmetic being another. In short, they were trying to come up with an axiomatic set theory. Cantor and Frege were doing a lot of this work, and made good progress using everyday language. They said that a set is any definable collection of elements, where “definable” means to provide a comprehension (a term you’re familiar with if you program in Python), or rule by which the set is constructed.

But along came Bertrand Russell. He pointed out a big problem in Cantor and Frege’s work, which is now called Russell’s paradox. Essentially, he made the following argument:

Y’all are saying any definable collection is a set. Well, how about this set: R, the set of all sets not contained within themselves. This is, according to you, a valid set, because I gave that comprehension. Now, R is not contained within itself, naturally: if it is contained within itself, then it being an element is a violation of my construction of R in the first place. But R must be contained within itself: if it’s not an element of itself, then it is a set that does not contain itself, and therefore it is an element of itself. So we have that R ∈ R and also R ∉ R. This is a contradiction! Obviously, your theory is seriously messed up.

This paradox is inherently a part of Cantor and Frege’s set theory — it shows that their system was inconsistent (with itself). As Qiaochu Yuan explains over at Quora, the problem is exactly what Russell pointed out: unrestricted comprehension — the idea that you can get away with defining any set you like simply by giving a comprehension. Zermelo and Fraenkel then came along and offered up a system of axioms that formalizes Cantor and Frege’s work logically, and restricts comprehension. This is called Zermelo-Fraenkel set theory (or ZF), and it is consistent (with itself). Cantor and Frege’s work was then retroactively called naive set theory, because it was, of course, pretty childish:

There are two more things worth knowing about axiomatic systems in mathematics. First, some people combine Zermelo-Fraenkel set theory with the Axiom of Choice¹, resulting in a set theory called ZFC. This is widely used as a standard by mathematicians today. Second, Gödel proved in 1931 that no system of axioms for arithmetic can be both consistent and complete — in every consistent axiomatization, there are “true” statements that are unprovable. Or put another way: in every consistent axiomatic system, there are statements which you can neither prove nor disprove. For example, in ZF, the Axiom of Choice is unprovable — you can’t prove it from the axioms in ZF. And in both ZF and ZFC, the continuum hypothesis² is unprovable.³ Gödel’s result is called the incompleteness theorem, and it’s a little depressing, because it means you can’t have any good logical basis for all of mathematics (but don’t tell anyone that, or we might all be out of a job). Luckily, ZF or ZFC has been good enough for virtually all of the mathematics we as a species have done so far!

The joke is that, when confronted with Russell’s paradox in naive set theory, the mathematician despairs, and wishes he could use Zermelo-Fraenkel set theory instead — ‘that feel when no ZF.’

I thought the joke was incredibly funny, specifically because of the reference to ‘tfw no gf’ and the implication that mathematicians romanticize ZF (which we totally do). I’ve definitely borrowed the joke to impress friends and faculty in the math department…a sort of fringe benefit of having a math blog.

– CJH

Keep reading

[Scene: the Nefarious Citadel. Minions are working busily after their eclipse-viewing half-day. Nefarious Asexual clears her throat over the PA system. The minions drop their tools and stand at attention.]

Greetings, dear minions! I hope you all enjoyed the eclipse!

I’ve received a lot of questions about the purpose of the Really Big Black Sack I’ve had you weaving in the past months. Remember that? The sack that covered the town and blighted all the crops with its merciless shadow? Of course you do. Well, this isn’t easy to say but in the interest of transparency…

That sack was supposed to be for stealing the sun.

I know now that my scheme to fold the sack up into a non-Euclidean space pocket, fly on a rocket to the sun, and chuck the sack onto the sun while the previously-stolen moon hid me from the roving eyes of the world was….ill-considered. I was drunk on the idea of purloining a full set of celestial bodies, and in my impulsive greed I thought it would be best to get the sun first and have all of the other planets fall in line in submission to me, the newly-crowned Asexual Sun Lord. The Nefarious Henchgirlfriend warned me that the sun was, quote,

“really big,”

but I was arrogant. How could the sun be bigger than my ambition and notorious asexual ego?

Really, really easily, as it turns out. 

Don’t fret, dear minions. Your labors will not be in vain. I’ll find a use for the Really Big But It Turns Out Not Nearly As Big As the Sun Black Sack, and pull off our grandest heist yet. The world will bow before us! It’s only a matter of time! HahahahahahHAHAHAHAHAHA!!!

Oh, and minion #2654, you left your headlights on.

Quantum Physics

Visualisation of Hilbert space

Cont’d from “Hilbert space”, see “Dirac notation: Analogy with Cartesian vectors

Now, in classical mechanics we deal with a vector space called Euclidean space (or Cartesian space), which is defined by the unitary vector components ex, ey and ez extending in a particular direction to infinity. Therefore this space is entirely filled by the vector e.

Hilbert space is, by contrast, a much more generalised and flexible version of this space. In Hilbert space, an “axis” can be any function or vector (not only e), which can extend for any distance or to infinity.

Imagine a space containing a number of representations completely orthogonal (i.e. at π∕2 radians, or 90°) to one another which are each represented by  a different basis function or vector. Each can have vectors projected on to them to reveal a different property of a corresponding vector or function. Suppose we have the basis functions ψi for i = 1 to 7 in this Hilbert space.

Notice how each basis vector is linearly dependent of every other basis vector within its set, meaning they are orthogonal and have an inner product of 0 with respect to one another.

Let’s now introduce into it a vector ψ, which has projections onto (or “components in”) each of the basis vectors ψi given by

As we can see, we’ve introduced this new vector in red and ‘visualised’ projecting it onto each of its basis vectors. We’ve previously discussed the similarities between Euclidean space and Hilbert space and can here draw a parallel between the process of resolving vectors on to their Cartesian axes and the projection of a state onto its basis.

Sofa Space

“Jay”

“Oh have you noticed the niftification i did to the sofa Charlie?”

“I certainly noticed something, it looks like it sags on one side, but if you look carefully it actually dissapears, where is the left hand bit of the sofa, Jay?”

“It’s there! on the other end!

"The left hand end of the sofa is on the right hand end! I might be getting a head ache here Jay, can we start with the why before we get to the how?”

“Well, last night you said the sofa was too small, but this room isn’t big enough, so i did a bit of thinkifying!”

“Thinkifying?”

“Yes! like a Jay! So in space yesterday i found a euclidian space! Which was a bit sad face as it had no friends! So i totally gave it a big hugging, and it showed me the model it made!”

“A model in euclidean space? this headache isn’t going to go is it?”

“Yes, a model of a nifty thing called a Möbius strip, which is smart like a Jay, it looks small, but has a never ending side!”

“So, where does the sofa fit in to thi…… Oh”

Jay, did you make a Möbius sofa?“

"Yes! i totally fixified the sofa to have a never ending side!”

“Oh, i bet that’s just Jaysome is it”

“Yes!

"Well, i think i shall put my feet up, on the bed, the Magician can try the sofa first”

KAL’L, THE BEAST FROM KRYPTON!

( @davidmann95 this is just some random ideas for Kal’l)

In the space between spaces, there was a dying realm known as KRYPTON! Older than age, beyond the reach of starlight, the denizens of this realm understood the inner workings of all reality. Until one day, their dreaming god Rao began to wake, making all of the space fall apart. The last surviving member of the race cast its child through the thinning veil of reality, depositing his egg in the garden of Martha Kent.

Krypto IS a squid, but a squid that drank some of his blood during a brutal undersea fight and gained his powers (and eldritch nature), so he started taking care of it.

The Fortress? on the moon. He doesn’t breath the same way as anyone else, drawing “Aether” from seemingly nowhere, so Kal’l made a den on the dark side of the moon.

A.A. Luthor, acclaimed cosmic horror writer, starts a society to fight back against the “horrors” after Kal’l accidentally gives his shut in mother a heart attack when she sees him on the news.

Instead of Mxyzptlk, he gets Nyarlathotep!

Giant Clay Juggernaut Diana and Kal’l still had a relationship, but it was ill-fated for some much grosser reasons. 

Instead of a cape, Kal’l wears a long dark red robe to keep his horrifying body from hurting anyone’s minds.

His “X-ray” vision is actually him peering into a non-euclidean space, and seeing the object from multiple impossible angles. 

Can eat almost anything, but doesn’t like to.

His version of Kryptonite is normal radioactive material (the energies react strangely to his interdimensional biology) and his Red Sun Radiation is magics that call upon his home realm, which can “Normalize” the space around him and make him JUST a cthulhu monster.

EDIT: ALSO THE FLASH IS AN EMPTY COSTUME HAUNTED BY THE SOULS OF PEOPLE WHO WENT TOO FAST AND EXITED REALITY

The hyperboloid

“In geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. The most familiar examples (illustrated here in three-dimensional Euclidean space) are the plane and the curved surface of acylinder or cone. Other examples are a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space.” - wikipedia

A ruled surface can always be described (at least locally) as the set of points swept by a moving straight line.

Girl at the Desk (1934). Kuzma Petrov-Vodkin (Russian, 1878-1939). Oil on canvas. Astrakhan State Picture Gallery.

Until the mid-1960s, Petrov-Vodkin was nearly forgotten in the Soviet Union after his curtailment of painting and turn towards writing. His most famous literary works are the 3 self-illustrated autobiographical novellas: “Khlynovsk,” “Euclidean Space” and “Samarcandia.” The second of these is of particular importance, as it transmits Petrov-Vodkin worldview as an artist in great detail.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as 6th Century BC.  By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.  Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. 

For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one absolutegeometry. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann’s new idea of space proved crucial in Einstein‘s general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.

 Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry.

The above image contains examples of simple 2D geometric shapes.

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.
—  Wikipedia

As a seemingly worrying aside, I’m having medical problems again. That’s business as usual for me, so no worries. The main thing is that I’m the world’s most annoying patient because whenever I’m around medical equipment I revert to a sugar-high 5-year-old. I can’t help it. Do you have ANY IDEA how much MATH is ALL OVER in medical facilities?

Example from this morning’s CT scan:

Me: “How does it work?”

CT Technician: *Explains*

Me: “Oh! So…” *Explains Lebesgue integration on euclidean space* “I’m thinking Tomography is an example of applied Lebesgue integration, because I was reading about how it takes cross sections in intervals and…”

CT Technician: “Ma’am, you can’t talk in the scan. You have to take a breath and hold it.”

Later:

Me: “Can I see it? Isn’t it weird that that’s my organs right there? Isn’t it nuts that we can do that with physics and mathematics, look inside a person without cutting them open? What do you think they’ll be able to do in the near future, like do you think surgery will become a thing of the past entirely?”

ok so I want to point out a neat geometrical fact and I don’t feel like drawing these things so I’ll just use some Wikimedia images

  • now we can replace the hexagons with pentagons (leaving the squares and triangles alone), but obviously there’s too much room in the Euclidean plane for that pattern to work, so we go to a place where there’s less room for stuff: spherical 2-space (aka the sphere), which we can see is completely tiled by this pattern, like this:

  • we can also try to replace the hexagons with heptagons, and clearly the Euclidean plane doesn’t have enough room for that, so it turns out that this variant tiles hyperbolic 2-space (where there’s more room), here presented in the form of the Poincaré disk, like this: