Background[ edit ] Euclidean geometrynamed after the Greek mathematician Euclidincludes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.
In implicit agreement with this, people often think that scale physically doesn't make any difference. The solar system could be an atom Non euclidean geometry some larger kind of matter.
Atoms could be little solar systems -- theories we see pondered in, of all things, the movie Animal House . In Men in Black , we see an entire galaxy Non euclidean geometry in a piece of jewelry, or for our own a marble.
The Fifty-Foot Woman [, ] may be big and slow, but otherwise her body works like that of an ordinary sized woman. We also get the franchise that began with Honey, I Shrunk the Kids  where scale didn't matter in the functioning of tiny human bodies. However, these are all physical impossibilities.
Atoms and solar systems work very differently. A galaxy small enough to be in a marble would be imploding into a Black Hole. Once the kids are shrunk, they would rapidly lose body heat and die of hypothermia -- small mammals like mice have elevated metabolisms that compensate.
Meanwhile, the metabolism of the Fifty-Foot Woman would be too much for her size, and she would experience hyperthermia and heat prostration. Elephants compensate with a slow metabolism.
But for there to be physical differences of scale, as occur in these cases, there must be a physical difference between different volumes of space. This means that space as such must be a physical thing, just as Newton or Kant, but not Leibniz, would have thought.
Scale and size alone make for physical differences. There is an absolute and a relative sense in which this is true. The absolute sense is that the Laws of Nature only operate above the scale of the Planck Lengthwhich is 4.
The relative sense of scale is in terms of density. A cubic meter of water, if contained, will simply sit there.
A cube of water, however, that was the radius of the Earth's orbit on a side would immediately undergo gravitational collapse and become a very, very massive star. A cubic meter of water dispersed in a cube that was the radius of the Earth's orbit an Astronomical Unit on a side would again be inactive, and in fact hardly detectable.
Density is the ratio of mass to volume. Without the physical reality of space, mass would be the only physical factor; and, according to Leibniz, the physical characteristics of the mass would only be a function of their relative arrangement.
The actual volume, however, changes all that. The mass can be identical, and the relative arrangement unchanged, but the actual volume will make for very different densities. Scale is literally one of the differences, and a major one, between an elephant and a mouse.
With their differences in size and mass, they simply cannot have the same structure. Thus, if we looked at models of a mouse and of an elephant, we could estimate, from their structure alone, their absolute size.
That is not something that Leibniz's metaphysics can explain, even as every volume contains an equally infinite number of Monads. Similarly, if we floated in our cube at right an amoeba, or galaxies as we see hereits scale would become apparent. These are things that actually do not exist at the opposite extremes of scale, i.
Leibniz cannot be excused from involvement with this argument, for he himself said that, if all bodies in the universe doubled in size overnight, we would notice no difference the next morning [cf.
This is an ambiguous challenge, since it might mean that the linear dimensions of all bodies would double, or that the actual volume of all bodies would double.
For such a thought experiment, that difference doesn't make a real difference, since the bodies will be significantly larger in either case. On the other hand, a correspondent has pointed out that the proposal is also ambiguous about what happens to the mass. Is the mass conserved and thus will remain the same as the body doubles in size or dimension?
Or are we talking about a body that increases proportionally in mass also? Of course, changing the mass at all contradicts the premise that we are only talking about space and its relative size.
But either way, whether we conserve the mass or scale it up, the result would be both noticeable and dangerous, since a larger body of the same mass or a larger body with scaled up mass will be physically different in either case from the smaller body.Pythagoras himself came up with the theory that numbers are of great importance for understanding the natural world, and he studied the role of numbers in music.
Although the Pythagorean theorem bears his name, the discoveries of the Pythagorean theorem and that the square root of 2 is an irrational number were most likely made after his death by his followers.
A reissue of Professor Coxeter's classic text on non-Euclidean geometry. It surveys real projective geometry, and elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases/5(5).
This survey of topics in Non-Euclidean Geometry is chock-full of colorful diagrams sure to delight mathematically inclined babies.
Non-Euclidean Geometry for Babies is intended to introduce babies to the basics of Euclid's Geometry, and supposes that the so-called "Parallel Postulate" might not be true.. Mathematician Fred Carlson believes that it's never too early to introduce children, and. Index for Geometry Math terminology from plane and solid geometry.
This includes basic triangle trigonometry as well as a few facts not traditionally taught in basic geometry. EGUMPP is the best online application for teaching grammar, usage, punctuation, and writing mechanics.
Improve your students grammar and writing skills with EGUMPP! Later in life, Gauss also claimed to have investigated a kind of non-Euclidean geometry using curved space but, unwilling to court controversy, he decided not to pursue or publish any of these avant-garde ideas.
This left the field open for János Bolyai and Nikolai Lobachevsky (respectively, a Hungarian and a Russian) who both independently explored the potential of hyperbolic geometry and.