clifford_a-_pickover_surfing_through_hyperspacebookfi-org
clifford_a-_pickover_surfing_through_hyperspacebookfi-org
clifford_a-_pickover_surfing_through_hyperspacebookfi-org
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DEGREES OF FREEDOM<br />
-9<br />
1800s. However, the philosopher Immanuel Kant (1724—1804) considered<br />
some of the spiritual aspects of a fourth dimension:<br />
A science of all these possible kinds of space would undoubtedly be<br />
the highest enterprise which a finite understanding could undertake<br />
in the field of geometry. ... If it is possible that there could be<br />
regions with other dimensions, it is very likely that a God had<br />
somewhere brought them into being. Such higher spaces would not<br />
belong to our world, but form separate worlds.<br />
Euclid (c. 300 B.C.), a prominent mathematician of Greco-Roman antiquity,<br />
understood that a point has no dimension at all. A line has one dimension:<br />
length. A plane had two dimensions. A solid had three dimensions. But<br />
there he stopped—believing nothing could have four dimensions. The Greek<br />
philosopher Aristotle (384—322 B.C.) echoed these beliefs in On Heaven:<br />
The line has magnitude in one way, the plane in two ways, and the<br />
solid in three ways, and beyond these there is no other magnitude<br />
because the three are all.<br />
Aristotle used the argument of perpendiculars to prove the impossibility of a<br />
fourth dimension. First he drew three mutually perpendicular lines, such as<br />
you might see in the corner of a cube. He then put forth the challenge to his<br />
colleagues to draw a fourth line perpendicular to the first three. Since there was<br />
no way to make four mutually perpendicular lines, he reasoned that the fourth<br />
dimension is impossible.<br />
It seems that the idea of a fourth dimension sometimes made philosophers<br />
and mathematicians a little nervous. John Wallis (1616—1703)—the most<br />
famous English mathematician before Isaac Newton and best known for his<br />
contributions to calculus's origin—called the fourth dimension a "monster in<br />
nature, less possible than a Chimera or Centaure." He wrote, "Length,<br />
Breadth, and Thickness, take up the whole of Space. Nor can fansie imagine<br />
how there should be a Fourth Local Dimension beyond these three."<br />
Similarly, <strong>through</strong>out history, mathematicians have called novel geometrical<br />
ideas "pathological" or "monstrous." Physicist Freeman Dyson recognized this<br />
for fractals, intricate structures that today have revolutionized mathematics<br />
and physics but in the past were treated with trepidation: