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### Einstein and Math

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Date: Sat, 4 Mar 1995 23:21:46 -0500 (EST)
From: Dov Peretz Elkins
Subject: Geometry

Hello, My name is Jonathan Stadlin.  I go to Princeton High school.
I have to give a presentation to my class on Albert Einstein and
I wondered if you could give me some information on how he related
to math or geometry.

A fellow student asked me if you could give him some information on
a mathematician named Sonya Kovaleski.

Thank you very much.
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Date: 4 Mar 1995 23:21:59 -0500
From: Dr. Ken
Subject: Re: Geometry

Hello there!

Yeah, the ideas that Einstein used in his work on redefining
the geometry of the universe are pretty neat.  To understand
them, you have to know a little bit about non-euclidean
geometry.

The geometry we all learn in high school is the geometry
of a plane, where there is only one line through a given
point that is parallel to another given line.  But it turns out
you can define two other perfectly good geometries:  where
given any line and a point not on that line, there is no line
through the given point that's parallel to the given line; and
where given any line and any point not on that line, there
are infinitely many lines through that point parallel to the
given line.  The first kind of geometry is called projective
geometry, and the second kind is hyperbolic geometry.

While this stuff is really neat, the important stuff is this:
if you study what's called the "curvature" of the surfaces
that you use for each of these geometries (this is a concept
that's pretty deep, and it can take up an entire college
course), you find that projective geometries have positive
curvature, euclidean geometry has zero curvature, and
hyperbolic geometry has negative curvature.  This, as I
understand, is the main idea that Einstein latched onto
when forming his theories.

Except for one key difference: when we do geometry in two
dimensions, well, we're in two dimensions.  So you have to
think about extending this concept of curving space to three
dimensions.

What Einstein said was that when an object sits in space,
it curves it a little bit, the same way a person sitting in the
middle of a trampoline curves the surface of the trampoline.
This is how you can think of the attraction of gravity:  two
objects are attracted to each other because the space in
between them is curved.  For instance, if you're sitting in the
middle of the trampoline and your sister throws a tennis ball
onto the trampoline, it will settle down into the depressed
region created by your weight.  That's like an object being
attracted to a big planet.

Of course, I'm not really an expert on Einstein's theories.  If
you really want to know the ins and outs of his work, you
could ask a Physics teacher at your high school, or better yet,
go to college and take some physics classes.  Neat stuff.

of Kovalevsky in my math history book, but it did say that she
did some good work in the theory of Differential Equations,
and she won the Paris Academy's prize for "a work of 1888 on
the integration of the equations of motion for a solid body
rotation around a fixed point; in 1889 she became a professor
of mathematics in Stockholm."  This is from Morris Kline's
book "Mathematical Thought from Ancient to Modern Times."

Here are a couple of biographies I turned up on her, too.  I hope
they help some.

AUTHOR       Kovalevskaia, S. V. (Sofia Vasilevna), 1850-1891.
TITLE        Sonya Kovalevsky.
PUBLISHER    N. Y., Century, 1895.
DESCRIPT     p. 155-297.
SUBJECT      Kovalevskaia, S. V. (Sofia Vasilevna), 1850-1891.
Autobiography.
Kovalevskaia, S. V. (Sofia Vasilevna), 1850-1891.
ALT. ENTRY   Hapgood, Isabel Florence, tr.
Cajanello, Anna Carlotta Leffler Edgren, duchessa di,
1849-1892.
Wolffsohn, Lily.
Bayley, A. M. Clive, tr.
IN           Kovalevskaia, Sof'ia Vasil'evna. Sonya Kovalevsky. 1895. p.
155-297.

AUTHOR       Cajanello, Anna Carlotta Leffler Edgren, duchessa di, 1849-1892.
TITLE        Sonya Kovalevsky : biography / by Anna Carlotta Leffler,
Duchess of Cajanello ; and, Sisters Rajevsky : being an
account of her [own] life / by Sonya Kovalevsky ; translated
by A. de Furuhjelm and A.M. Clive Bayley ; with a biographical
note by Lily Wolffsohn.
EDITION      Authorized ed.
PUBLISHER    London : T. Fisher Unwin, 1895.
DESCRIPT     x, 377 p., 1 leaf of plates : port. ; 24 cm.
SUBJECT      Kovalevskaia, S. V. (Sofia Vasilevna), 1850-1891.
ALT. ENTRY   Bayley, Anna M.
Furuhjelm, A. de, tr.
Kovalevskaia, S. V. (Sofia Vasilevna), 1850-1891. Sisters
Rajevsky.
ALT. TITLE   The sisters Rajevsky.

-Ken "Dr." Math
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Associated Topics:
High School History/Biography

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