The Bible on the Fourth Dimension?
• “That Christ may dwell in your hearts by faith;
that ye, being rooted and grounded in love,
May be able to comprehend with all saints what
is the breadth, and length, and depth, and
height.” (Ephesians 3:17-18)
• “And he that talked with me had a golden reed
to measure the city … and the city lieth
foursquare …” (Revelation 21:15-16a)
• W. A. Granville, The Fourth Dimension and the
Bible, Gorham Press : Boston, 1922.
Jody Trout (Dartmouth College)
Mathematicians against the 4th-D!!
• Aristotle (in On Heaven) wrote, “The line has
magnitude in one way, the plane in two ways, and the
solid in three ways, and beyond these there is no
other magnitude because the three are all.”
• Ptolemy of Alexandria (150 A.D.) offered a “proof”
that the fourth dimension does not exist in his book
On Distance.
– Note that he also “proved” that the Earth was the
center of the Solar System…
– And “proved” that Euclid’s Parallel (Fifth) Postulate
followed from the others!
Jody Trout (Dartmouth College)
More mathematicians against the
Fourth Dimension!
• Nicole Oresme (1323 - 1382) first conceived of using what we
know call “rectangular coordinates” (longitudo and latitudo) but
rejected a “fourth dimension” (4am dimensionem).
• The Italian mathematician Cardan in Ars Magnus (1545) says
about the powers of numbers, “the first power refers to a line,
the square to a surface, the cube to a solid and that it would
be fatuous indeed for us to progress beyond, for the reason
that it is contrary to nature.”
• The German mathematician and astronomer Clavius (1538 1612) tried to improve on Ptolemy’s proof that no more than
three perpendicular lines can be drawn in space.
Jody Trout (Dartmouth College)
Henry More (1614 - 1687)
• First philosopher to seriously entertain the idea of a “fourthdimension” of space.
• In his book Immortality of the Soul (1659), he uses the term
spissitude to describe a fourth spatial dimension in which he
believed the spiritual realm extended. The term refers to a
measurement of an object's length along a direction in the
fourth spatial dimension, analogous to the three-dimensional
terms length, breadth, and height.
• Actually uses the term “fourth dimension” in his metaphysicial
text Enchiridion Metaphysicum (1671) where he says, “That
besides the three dimensions which are filled with all extended
material things, a fourth must be admitted, with which
coincides the spirit.”
Jody Trout (Dartmouth College)
But the mathematicians are still against the 4thD!!
John Wallis in his book Algebra (1685) says…
A Line drawn into a Line shall make a Plane or Surface; this
drawn into a Line, shall make a Solid: But if this Solid be
drawn into a Line, or this Plane into a Plane, what shall it
make? A Plano-plane? That is a Monster in Nature, and less
possible than a Chimaera or Centaure. For Length, Breadth
and Thickness, take up the whole of Space. Nor can our
Fansie imagine how there should be a Fourth Local
Dimension beyond these Three.
Whereas Nature does not admit of more than three
dimensions... it may justly seem very improper to talk of a
solid... drawn into a fourth, fifth, sixth, or further
dimension.
Jody Trout (Dartmouth College)
History of the Fourth Dimension
•
370 BC: Plato’s cave allegory of shadows on the wall used analogies between 2D
(shadows) and 3D objects. (Book VII of the Republic)
•
1659-1671: Henry More suggests that spirits should be four-dimensional (have
spissitude). First serious acceptance of a fourth dimension!
•
1747: Immanuel Kant speculates on the philosophical implications of higherdimensional spaces in his thesis.
•
1754: D’Alembert (and Lagrange in 1797) view time as a fourth dimension.
•
1827: August Möbius discovers that a 3-D object can be turned into it’s mirror
opposite by a 4-D rotation. First mathematical use of a fourth dimension!
•
1846: Gustav Fechner writes story “Space Has Four Dimensions” in Vier Paradoxe
under pseudonym “Dr. Mises”. Cayley uses four-dimensional geometry.
•
1851: Sylvester discusses tangent and polar forms in n-dimensional geometry.
•
1854: Bernhard Riemann (1854) develops the differential geometry for (curved)
spaces of any dimension in his doctoral thesis The Hypotheses Which Underlie
Geometry.
•
1876: Trial of Henry Slade sparks interest in the “notorious” 4th Dimension.
•
1880: Charles H. Hinton writes influential article “What is the Fourth Dimension?”
in the Cheltenham Ladies College Magazine and Dublin University Magazine.
Jody Trout (Dartmouth College)
Immanuel Kant (1724-1804)
• Also philosophized about the idea of “higherdimensional” spaces.
• “A science of all these possible kinds of space
would undoubtedly be the highest enterprise which
a finite understanding could undertake in the field of
geometry…If it is possible there could be regions
with other dimensions, it is very likely that God has
somewhere brought them into being.”
-- Thoughts on the True Estimation of Living Forces (1747)
Jody Trout (Dartmouth College)
Mystics and Mathematicians
• In the late 1800s - early 1900s, the fourth dimension
was used to explain certain mystical phenomena:
 Ghosts ability to travel through walls.
 Spirits ability to remove objects from locked rooms.
 Heaven and Hell are in the Fourth Dimension!
• In 1876, American psychic Henry Slade was put on trial
for séance fraud in Britain. Curiously, famous scientists
came to his defense and used the Fourth Dimension to
explain his mysterious “powers”:
o J.H.C. Zöllner (Univ. of Leipzig) organized the defense
o William Crookes - inventor of the Cathode Ray Tube
o Wilhelm Weber - unit of magnetism, worked with Gauss
o J.J. Thompson - discoverer of the electron (Nobel 1906)
o Lord Rayleigh - discoverer of argon (Nobel 1904)
Jody Trout (Dartmouth College)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
E. A. Abbott, Flatland (1884)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Jody Trout (Dartmouth College)
Recounts A. Square’s
journey into higher
dimensions. Also
gives a stinging
satire on Victorian
society and the
treatment of women.
Note: Abbott was not
a mathematician!
Scenes from Flatland
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
The Golden 4D Years: 1890-1910
The fourth dimension appeared in the literary works of Oscar
Wilde, Fyodor Dostoyevsky, Marcel Proust, H. G. Wells, and
Joseph Conrad, and inspired some of the musical works of
Alexander Scriabin, Edgard Varese, and George Antheil. It
fascinated many people: psychologist William James, literary
figure Gertrude Stein, revolutionary Vladimir Lenin, inspired
the artistic works of Pablo Picasso and Marchel Duchamp,
and also Einstein’s theories of relativity, of course!
-- M. Kaku, Hyperspace.
“The fourth dimension became a thing you talked about,
without knowing what it meant."
-- Marcel Duchamp
However, by 1911, mathematicians had written over 1830
research articles and books on n-dimensional geometry!
Jody Trout (Dartmouth College)
Marcel Duchamp (1912)
Nude Descending a Staircase
“The shadow cast by a fourdimensional figure on our space is
a three-dimensional shadow…by
analogy with the method by which
architects depict the plan of each
story of a house, a fourdimensional figure can be
represented (in each one of its
stories) by three-dimensional
sections. These different sections
will be bound to one another by
the fourth dimension.”
-- Marcel Duchamp
Jody Trout (Dartmouth College)
C. Howard Hinton: The Man Who Saw
The Fourth Dimension
• In 1888, suggested
that space had a
slight 4-D thickness.
• Created a technique
for seeing 4-D using
a set of 27 colored
cubes.
Jody Trout (Dartmouth College)
• In 1893, fired from
Princeton….
The Necker Cube Illusion
• Which is the “front” face of
the cube?
• While staring at the cube, it
starts to “flip”.
• This “flipping” is
mathematically equivalent
to a rotation through the
fourth dimension!
• 3D reflection thru plane =
4D rotation matrix! (Take a
linear algebra class…)
Jody Trout (Dartmouth College)
Jody Trout (Dartmouth College)