The only way to finish college
The only way to finish college
now that’s how you use mathematics
why sun and the moon appear to be the same size?
IMPOSSIBLE! Right? You may have heard “the interior angles of a triangle always add up to 180 degrees”. This is not always true. Check out the second image, it shows a triangle with 3 right angles for a total of 270 degrees!
It is true in flat Euclidean geometry (the geometry you probably learned in school) however. But there are so many other geometries out there! You may be thinking, are other geometries real though? A mathematician would argue they are just as real as the typical flat geometry you know and love (or hate). These alternative geometries can be practically useful too!
The images above show triangles in spherical geometry. Those aren’t triangles though! Oh but they are! A triangle is just a polygon enclosed by three lines. Looks like it fits the criteria. Wait but those aren’t lines, they are curved! Ah yes. I argue that these are, for all intents and purposes, just as good as lines. We need to ask: What is a line? A line is so basic to us we may not know how to describe it. I offer this definition: A line is the shortest path between 2 points. The 3 curves that make the triangle above are in fact the shortest paths from one vertex to the other on the surface of the sphere (they just so happen to be on circumferences of the sphere, which are often referred to as great circles). So it may be more useful to think of lines, in general, as length minimizing curves. In conclusion, we would consider the shape above to be a triangle as it is enclosed by 3 length minimizing curves on a surface.
Spherical geometry can be very useful; think about the Earth. To reduce travel time, airplanes would want to travel along great circles as they are the shortest paths from one place to another. Additionally, this type of thinking (rethinking straight lines as length minimizing curves) is central to Albert Einstein’s general theory of relativity.
read more at http://staffrm.io/@missnorledge/35H6cS1T52
the limits of the universe
“Radio Power Will Revolutionize The World.” By Alfred Albelli. Modern Mechanix, July, 1934.
If one remembers this particular episode from the popular sitcom ‘Friends’ where Ross is trying to carry a sofa to his apartment, it seems that moving a sofa up the stairs is ridiculously hard.
But life shouldn’t be that hard now should it?
The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what
is the shape of largest area in the plane that can be moved around a
right-angled corner in a two-dimensional hallway of width 1? This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked.
The most common shape to move around a tight right angled corner is a square.
And another common shape that would satisfy this criterion is a semi-circle.
what is the largest area that can be moved around?
Well, it has been
conjectured that the shape with the largest area that one can move around a corner is known as “Gerver’s
sofa”. And it looks like so:
Wait.. Hang on a second
sofa would only be effective for right handed turns. One can clearly
see that if we have to turn left somewhere we would be kind of in a tough
Prof.Romik from the University of California, Davis has
proposed this shape popularly know as Romik’s ambidextrous sofa that
solves this problem.
Although Prof.Romik’s sofa may/may not be the not the optimal solution, it is definitely is a breakthrough since this can pave the way for more complex ideas in mathematical analysis and more importantly sofa design.
Have a good one!
“…Mr. Tesla was charmed to hear about the Vedantic Prâna and Âkâshâ and the Kalpas, which according to him are the only theories modern science can entertain. Now both Âkâshâ and Prâna again are produced from the cosmic Mahat, the Universal Mind, the Brahmâ or Ishvara. Mr. Tesla thinks he can demonstrate mathematically that force and matter are reducible to potential energy. I am to go and see him next week, to get this new mathematical demonstration.
“In that case, the Vedantic cosmology will be placed on the surest of foundations. I am working a good deal now upon the cosmology and eschatology (That is, doctrine of the last things — death, judgement, etc.) of the Vedanta. I clearly see their perfect unison with modern science, and the elucidation of the one will be followed by that of the other. I intend to write a book later on in the form of questions and answers. (This was never done. But from his lectures in London in 1896, it is easy to see that his mind was still working on these ideas. (See also Vol. VIII Sayings and Utterances& Letter to Mr. Sturdy .)). The first chapter will be on cosmology, showing the harmony between Vedantic theories and modern science.
Brahmâ = The Absolute
Mahat Ishwara = Primal Creative Energy
Prâna and Âkâshâ = Force and Matter
“The eschatology will be explained from the Advaitic standpoint only. That is to say, the dualist claims that the soul after death passes on to the Solar sphere, thence to the Lunar sphere, thence to the Electric sphere. Thence he is accompanied by a Purusha to Brahmaloka. (Thence, says the Advaitist, he goes to Nirvâna.)
“Now on the Advaitic side, it is held that the soul neither comes nor goes, and that all these spheres or layers of the universe are only so many varying products of Âkâshâ and Prâna. That is to say, the lowest or most condensed is the Solar sphere, consisting of the visible universe, in which Prana appears as physical force, and Âkâshâ as sensible matter. The next is called the Lunar sphere, which surrounds the Solar sphere. This is not the moon at all, but the habitation of the gods, that is to say, Prâna appears in it as psychic forces, and Akasha as Tanmâtras or fine particles. Beyond this is the Electric sphere, that is to say, a condition in which the Prâna is almost inseparable from Âkâshâ, and you can hardly tell whether Electricity is force or matter. Next is the Brahmaloka. where there is neither Prâna nor Âkâshâ, but both are merged in the mind stuff, the primal energy. And here — there big neither Prâna nor Âkâshâ — the Jiva contemplates the whole universe as Samashti or the sum total of Mahat or mind. This appears as a Purusha, an abstract universal soul, yet not the Absolute, for still there is multiplicity. From this the Jiva finds at last that Unity which is the end. Advaitism says that these are the visions which rise in succession before the Jiva, who himself neither goes nor comes, and that in the same way this present vision has been projected. The projection (Srishti) and dissolution must take place in the same order, only one means going backward, and the other coming out.
“Now as each individual can only see his own universe, that universe is created with his bondage and goes away with his liberation, although it remains for others who are in bondage. Now name and form constitute the universe. A wave in the ocean is a wave, only in so far as it is bound by name and form. If the wave subsides, it is the ocean, but those name and form have immediately vanished for ever. So though the name and form of wave could never be without water that was fashioned into the wave by them, yet the name and form themselves were not the wave. They die as soon as ever it returns to water. But other names and forms live in relation to other waves. This name-and-form is called Mâyâ, and the water is Brahman. The wave was nothing but water all the time, yet as a wave it had the name and form. Again this name and form cannot remain for one moment separated from the wave, although the wave as water can remain eternally separate from name and form. But because the name and form can never he separated, they can never be said to exist. Yet they are not zero. This is called Maya.
“I want to work; all this out carefully, but you will see at a glance that I am on the right track. It will take more study in physiology, on the relations between the higher and lower centres, to fill out the psychology of mind Chitta (mind-stuff), and Buddhi (intellect), and so on. But I have clear light now, free of all hocus-pocus. I want to give them dry, hard reason, softened in the sweetest syrup of love and made spicy with intense work, and cooked in the kitchen of Yoga, so that even a baby can easily digest it.”
(A Letter To Mr. E.T. Sturdy. 228 W. 39th Street, New York, February, 13, 1896.