"Benford-FourthDimension" - читать интересную книгу автора (Benford Gregory)

their views certainly do not embrace equality. Mathematicians term equal-sided
figures "regular," and in nineteenth century terms, proper upper class polygons
are of the regular sort.

A Square learns that his view of the world is too narrow. There is a third
dimension, grander and exciting. but his hidebound fellows cannot see it. This
opening-out is the central imaginative event of the novel, Abbott echoing an
emergent idea.

In the late nineteenth century higher dimensions were fashionable.
Mathematicians had laid the foundations for rigorous work in higher-dimensional
space, and physicists were about to begin using four-dimensional spacetime.
Twenty centuries after Euclid, the mathematician Bernhard Riemann took a great
leap in 1854, liberating the idea of dimensions from our spatial senses. He
argued that ever since Rene Descartes had described spaces with algebra, the
path to discussing higher dimensions had been dear, but unwalked.

Descartes' analytic geometry defined lines as things described by one set of
coordinates, distances along one axis. A plane needed two independent coordinate
sets, a solid took three. With coordinates one could map an object, defining it
quantitatively: not "Chicago is over that hill." but "Chicago is fifteen miles
that way." This appealed more to our logical capacity, and less to our sensory
experience.

Riemann described worlds of equal logical possibility, with dimensions ranging
from one to infinity. They were not spatial in the ordinary sense. Instead,
Riemann took dimension to refer to conceptual spaces, which he named manifolds.

This wasn't merely a semantic change. Weather, for example, depends on several
variables -- say, n -- like temperature, pressure, wind velocity, time of day,
etc. One could represent the weather as a moving point in an n-dimensional
space. A plausible model of everyday weather needs about a dozen variables, so
to visualize it means seeing curves and surfaces in a twelve-dimensional world.
No wonder we understand the motions of planets (which even Einstein only needed
four dimensions to describe), but not the weather.

Riemann revolutionized mathematics and his general ideas diffused into our
culture. By 1880, C.H. Hinton had pressed the issue by building elaborate models
to further his extra-dimensional intuition, he tried to explain ghosts as
higher-dimensional apparitions. Pursuing the analogy, he wrote of a
fourth-dimensional God from whom nothing could be hidden. The afterlife, then,
allowed spirits to move along the time dimension, reliving and reassessing
moments of life. Spirits from hyper-space were the subject of J.K.F. Zollner's
1878 Transcendental Physics, which envisioned them moving everywhere by
short-cut loops through the fourth dimension.

Mystics responded to the fashion by imagining that God, souls, angels and any
other theological beings resided as literal beings of mass ("hypermatter") in
four-space. This neatly explains why they can appear anywhere they like, and God
can be everywhere simultaneously, the way we can look down on a Flatland and