The Fourth Dimension and Non-Euclidean Geometry in Modern Art

[Dave Monroe]

>From Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean
Geometry in Modern Art (Princeton, NJ: Princeton UP, 1983), Ch. 1,
"Nineteenth-Century Background," pp. 3-43 ...


    In the end, the definition of the fourth dimension as time was
actually to displace popular interest in higher places.  Following its
use by H.G. Wells in his science fiction tale of 1895, The Time
Machine, a temporal fourth dimension became part of the science "fact"
of Minkowski's space-time continuum for Einstein's Theory of
relativity in 1908.  However, in the late nineteenth- and early
twentieth-century literature on the fourth dimension, time was always
the less important of the two interpretations of the fourth dimension.
 If, in certain more philosophical and mystical expositions of a
spatila fourth dimension, time played arole in the process of
visualizing a higher dimension of space, time itself was not
interpreted as the fourth dimension.  It was the geometry of higher
dimensions of space, along with non-Euclidean geometry, which
fascinated the public in the early twentieth century.

The Rise of Popular Interest in the New Geometries

   Ideas deriving from the new geometries, which had been the province
of mathematicians alone in the first half of the nineteenth century,
gradually began to appear in nonmathematical literature from the 1860s
onward.  This process is recorded in the listing sof popular articles
in Duncan Sommerville's Bibliography of Non-Euclidean Geometry,
Including ... Space of n Dimensions.  Two specific areas of
philosophical debate were the initial sources of public inetrest in
non-Euclidean geometry and the geometries of higher dimesions: the
nature of geometrical axioms and the nature of our space.  Controversy
about the nature of geometrical axioms naturally resulted from the
challenge non-Euclidean geometry posed to Kant's view that the axioms
of mathematics were a priori .... (pp. 9-10)


   The philosophical impact of non-Euclidean geometry in the
nineteenth century was far greater than simply its initial challenge
to Kant.  It substantially shook the foundations of mathematics and
science, branches of learning that for two thousand years had depended
on the truth of Euclid's axioms.  As a result, optimistic belief in
man's ability to acquire absolute truth gradually gave way during the
later nineteenth century to a recognition of the relativity of
knowledge.  Coming full circle from its early days as a tool of the
empiricist, positivist Helmholtz, non-Euclidean geometry contributed
substantially to the demise of traditional positivism.  For certain
artists in the early twentieth century, non-Euclidean geometry was to
be synonymous with the rejection of tradition and even with
revolution.

The Popularization of n-Dimenasional Geometry and the Fourth Dimension
in England and France

   From a survey of Sommervile's Bibliography of Non-Euclidean
Geometry, Including ... Space of n Dimensions, England in the 1870s
emerges as the first center of active concern with the number of
dimensions of space.  This decade of development culminated in 1884
with the publication of E.A. Abbott's Flatland .... (p.17)


Citing ...

Abbott, E.A.  Flatland: A Romance of Many Dimensions (1884)

http://www.ibiblio.org/eldritch/eaa/FL.HTM
http://www.gutenberg.org/ebooks/201
http://www.der.org/films/flatland.html
http://xahlee.org/flatland/index.html
http://en.wikipedia.org/wiki/Flatland

Sommerville, Duncan M.Y.  Bibliography of non-Euclidean Geometry,
   including the Theory of Parallels, the Foundations of Geometry and
   Space of n Dimensions.  London: Harrison and Sons, 1911.