search for books and compare prices
Tables of Contents for The Fourth Dimension
Chapter/Section Title
Page #
Page Count
PART I LECTURES ON THE FOURTH DIMENSION
About this Edition
xiii
Introduction
xvii
David Booth
First Lecture, Berlin, March 24, 1905
1
10
Mathematical thinking and reality
The dimensions of space
Movement as a means of passing from lower to higher dimensions
Mirror-image symmetry
Analogies for the relationship between the outer world and internal sensation: bending ever larger line segments into circles, seal and sealing wax
The fourth dimension as a conceptual possibility and as reality
Enlivening our visualization of space: Oscar Simony's models
Second Lecture, Berlin, March 31, 1905
11
8
Thoughts on four-dimensional space in connection with Hinton's work
Symmetrical relationships
``Loops'' in space as real natural processes and forces, for example, the movement of the Moon and Earth around the Sun
Building up the dimensions
Human beings as four-dimensional beings; at earlier stages of our evolution, we were three-dimensional
The astral world
Point and periphery; a point radiating light outward is the opposite of a sphere radiating darkness toward the center
The cube and its opposite
Radiant ability as an additional dimension; its application to squares and cubes
Third Lecture, Berlin, May 17, 1905
19
14
Studying four-dimensional space as preparation for understanding the astral world and other forms of higher existence
Characteristic attributes of the astral world: numbers, spatial figures, and time sequences must be read symmetrically, that is, in reverse or as their own mirror images
Morality also appears in inverted or mirror-image form
The periphery is the center
Human life as a backup in the two streams of time running from the past and the future
The threshold, as an astral experience of the panorama a future evolution, includes the question, Do you choose to enter? Kamaloka reveals the unpurified animal nature of the human being; this is the deeper meaning of the doctrine of transmigration of souls
A square on the physical and mental level
The physical square as a stoppage in two pairs of opposing streams
A cube on the physical and mental level
Positive and negative dimensions
The astral world is four-dimensional
The animal as a stoppage in the opposing streams represented by the plant and the human being
Fourth Lecture, Berlin, May 24, 1905
33
10
Exercises in representing three-dimensional figures in two dimensions, with reference to Hinton
Unfolding the three dimensions of a cube; using colors to represent them
Representing the third dimension in a plane by moving a bicolored square through a third color
Transferring this operation to the representation of a four dimensional figure, the tessaract
Unfolding a tessaract compared with unfolding a cube
The alchemical mystery and a true view of four-dimensional space
Meditative visualization of mercury, sulfur
Astral substance
Fifth Lecture, Berlin, May 31, 1905
43
12
Unfolding a cube as a new analogy for three-dimensional representation of a tessaract (four-dimensional cube)
Analogy as a method of developing a mental image of four-dimensional figures
Halving the number of surfaces in an octahedron produces a tetrahedron; this operation cannot be performed on a cube
The geometric properties of the rhombic dodecahedron compared with the cube and the tetrahedron/octahedron
The cube as the counterpart of three-dimensional space
Curved figures bounding two- and three-dimensional figures: curved squares and curved cubes
An ordinary cube is the result of flattening a curved cube
Reversing this operation and curving a three-dimensional figure can result in a four-dimensional figure
Sixth Lecture, Berlin, June 7, 1905
55
12
A hexagon is the projection of a cube into two-dimensional space; a rhombic dodecahedron is the projection of a tessaract into three-dimensional space
The axes of the cube and rhombic dodecahedron
Plato's cave metaphor as an image of the relationship between four-dimensional reality and three-dimensional space
Movement or time as the expression and manifestation of life, the fourth dimension
Crystals have planes for boundaries, while living things have spherical boundaries
Destroying a living thing's fourth dimension results in a static three-dimensional image
The fifth dimension, which results from encounters between four-dimensional beings, manifests in the third dimension as sensory activity
Self-awareness is the projection of the sixth dimension into the three-dimensional physical world
What Moses experienced on Mount Sinai is an example of a real four-dimensional being with two ordinary dimensions plus the two higher dimensions of time and sensory activity
The development of spiritual abilities through intensive inner work with the analogies presented here
Four-Dimensional Space Berlin, November 7, 1905
67
10
Creating dimensions through movement
Transforming a circle into a straight line
The importance of recent synthetic, projective geometry in considering space correctly
Space is self-contained
Twists in closed curves (strips of paper) as an example of intertwining dimensions
In reality, the movements of the Moon and Earth around the Sun are similarly intertwined
Enlivening our view of space
Unfolding the cube in a plane and the tessaract in three-dimensional space
A hexagon is a projection of a cube; a rhombic dodecahedron is a projection of a tessaract
The transition to reality
Time, movement, and development as expressions of the fourth dimension in plants
When time itself becomes alive, sensory ability emerges as the expression of the fifth dimension in animals
Human beings are six-dimensional beings
On Higher-Dimensional Space Berlin, October 22, 1908
77
10
Mathematicians can do no more than discuss the possibility of higher-dimensional space
The cube's three dimensions of length, width, and height
What is a plane surface? A merely arithmetical transition to higher dimensions does not lead to reality
Attempting to understand space--for example, infinity--on the basis of numbers leads to confusion
Numbers have no relationship or a neutral relationship to space
Observing repeated disappearances and reappearances points to the existence of a fourth dimension
Refuting a materialistic argument
Unfolding the boundaries of a square and a cube
Unfolding the eight cubes of a tessaract
PART II QUESTIONS AND ANSWERS 1906-1922
Stuttgart, September 2, 1906
87
2
Occult schooling means working on our astral body and ether body
The astral world is four dimensional
Life reveals its fourth dimension through growth
Comparison to an ever-widening circle that becomes a straight line
Astral space is self-contained
Nurnberg, June 28, 1908
89
1
Astral space, in contrast to physical space, is not limited
It behaves like a self-contained projective straight line
Visualizing a circle expanding to become a straight line
Dusseldorf, April 21, 1909
90
2
Space from the esoteric perspective
The relationship of the hierarchies and the Trinity to space
Time as a result of the interaction of lower and higher beings
Even for the hierarchies, space exists as a creation of the Trinity
Dusseldorf, April 22, 1909
92
1
Working with basic geometric concepts awakens clairvoyant abilities
A self-contained (projective) straight line as an example of astral spatial relationships
Berlin, November 2, 1910
93
1
Plants, animals, and human beings as four-, five-, and six-dimensional beings, respectively
Basel, October 1, 1911
94
1
Light possesses inwardness as the fourth dimension
Munich, November 25, 1912
94
2
The question of the reality of higher dimensions
Mathematicians can formulate theoretical ideas about these dimensions
Higher reality is indeed higher dimensional, but we would need better mathematics to do it justice
Some subjects on the fringes of mathematics are important
The example of a projective straight line
We must not overestimate mathematics
Berlin, February 13, 1913
96
1
The occult significance of the Golden Section
Berlin, November 27, 1913
97
1
In life after death, space and time are totally different; speed, rather than time, belongs to our inner experience
Time is dependent on processes of inner development
Stuttgart, 1919
98
1
Written answers to questions on mathematics
Stuttgart, March 7, 1920
98
7
The speed of light and the propagation of light waves
Mechanical measuring methods are not applicable to light
Light as it spreads outward is not lost in infinity but is subject to a law of elasticity
Problems regarding Einstein's theory of relativity from the perspective of spiritual science
In mechanics, time is an abstraction; only speed is real
Discussion about the formula for speed
The length of an organism's life and its size are not relative or arbitrary
The theory of relativity needs to be countered by a theory of the absolute character of total systems
Stuttgart, March 7, 1920
105
2
The energy stored in a mass (according to Einstein's theory) can be technologically utilized if it can be harnessed
Einstein's formula E = Mc2 is a type of potential energy
The problem of making arithmetical procedures absolute
The immanent time of total systems
Stuttgart, March 11, 1920
107
6
Positive and negative numbers as reality; ponderable and imponderable matter
Symbolizing the color spectrum
Positive numbers--physical reality; negative numbers--extraspatial, etheric reality; imaginary numbers--astral realm; superimaginary numbers--the true being of the ``I''
Zero factors must be included
The human being as a state of balance between the suprasensible and the subsensible
Number systems on curved surfaces
The concept of ``only calculable'' in mathematics
We need to be able to conceive of negative and imaginary numbers without the help of geometry
Stuttgart, March 11, 1920
113
5
The realms of mathematics and geometry are intermediary states between an archetype and its image on the physical plane
Inherently mobile or fluid geometry
Higher dimensions
The human being as a physical image of spiritual worlds
Color perspective
Expanding fluid geometry by means of an intensity factor with the help of colors
Stereoscopic vision as the balanced interaction of the left and right eyes; this enlivened vision is the dynamic center of asymmetrical organs
Dornach, March, 30, 1920
118
2
Phenomenology as the systematization of phenomena
The relationship of an axiom to geometric correlations is comparable to the relationship of an archetypal phenomenon to derivative phenomena
The need to clarify the concept of experience
The discovery of non-Euclidean geometry shows that mathematical formulas, like phenomenological conclusions, require empirical verification in reality
Dornach, March 31, 1920
120
7
Extensions of mathematics
Real phenomenology is concerned with beings, but the urge for mechanical control excludes beings and leads to many technical accomplishments at the expense of the progress of real knowledge, that is, knowledge of the human being
Goethe's color theory
Expanding our perspective also requires expanding the field of mathematics
The ether is not to be imagined in material terms
When we enter the etheric realm, we must substitute negative numbers for positive numbers in mathematical formulas
Transcending the domain of life requires the substitution of imaginary numbers, which could offer a way out of our present plight, that is out of our merely technological control of nature
Dornach, October 15, 1920
127
9
Copernicus's third law is unjustifiably disregarded
In reality, the Sun moves along a spiral line, and the Earth and the other planets follow it
Science must include the human being if it is to reflect reality
The theory of relativity leads to abstractions
Space and time are abstractions; only speed is real
Copernicus's third law and Bessel's corrections
Mathematical thinking that lacks a sense for reality leads to unreality
In set theory, number is dissolved, and we swim in abstractions
In The Decline of the West, Oswald Spengler courageously develops consistent concepts based on reality, but these concepts do not fit together
Herman Keyserling offers only empty husks of words
Stuttgart, January 15, 1921
136
2
The study of phenomena as the basis of the anthroposophical broadening of knowledge
Mathematical formulas must be verified on the basis of reality
The theory of heat
Einstein's theory is based on thought experiments
Conductive and radiant heat as positive and negative
The radial and peripheral direction of the effect must be added
The anthroposophical position does not precede phenomena but is appropriately based on them
In future, we need an enhancement of the truly scientific approach
Dornach, April 7, 1921
138
7
In mathematics, the dimensions of space are interchangeable
The need to distinguish between boundedness and infinity (Riemann)
The concepts of metageometry (non-Euclidean geometry); Gauss
Mathematical space--either Euclidean space or the space of Riemann's or other geometries--is abstract
Kant's view of space is shattered by mathematics
The derivatives of modern metageometry contain a circular argument
To achieve a concept of space that corresponds to reality, we must begin with human experience
Derivation of the dimension of depth, which is not interchangeable with any other dimension
Imagination leads to two-dimensional visualization, inspiration to one-dimensional visualization
In real space dimensions are not interchangeable; there are different intensities in different directions
Fixed space is an abstraction derived from real space
The theory of relativity is logical but foreign to reality
Dornach, August 26, 1921
145
3
A brief sketch of the results of spiritual scientific research on the spiral movements of Earth and Sun
The conclusions of most models of the solar system are one-sided and result from very specific perspectives
The Sun moves in a spiral path, and the Earth follows it
In reality, only the direction of our view from the Earth to the Sun turns
All other movements are much more complicated
Copernicus's third law has been overlooked
The Hague, April 12, 1922
148
11
Extending the coordinate axis system on an abstract level leads to four-, five-, and ultimately n-dimensional spaces
Hinton and the tessarct
Time as the fourth dimension is based on an abstract understanding of space
The fourth dimension actually negates the third, so that only two dimensions remain
Similarly, the fifth dimension negates the second, so we revert to one dimension
To explain the shape of a flower, we must set the beginning point of the coordinates in an infinitely large sphere and move centripetally inward
In the etheric realm, gliding and scraping movements appear
The hyperbola as an example
Through synthetic geometry, we gradually develop a concrete, reality-based way of handling space
Einstein's theory of relativity is absolutely correct and irrefutable with regard to three-dimensional, perceived space
It looks different when we make the transition to the etheric domain
The ether body inhabits total space
Through inner vision, we arrive at absolutes
The theory of relativity evaluates everything from the perspective of the observer and cannot be refuted here from that perspective
Its validity ceases, however, when we enter the spiritual realm, where boundaries between object and subject also are eliminated
To understand the physical body as a spatial body and the body of formative forces as a temporal body, we must separate the concepts of space and time
Time usually is measured only in spatial units
That is not the case in the true experience of time that appears with imaginative vision
At a certain point in a human lifetime, a temporal cross-section of the soul's life occurs
This cross-section includes the person's entire earthly past
Perspectives that depend on our soul life
``Later'' and ``earlier'' are organically, not superficially related, like spatial relationships
Folded hands in youth become blessing hands in old age
The time organism is revealed fully only to Imagination, but we can gain an idea of it by studying temporal processes in our soul life
Ostwald says that organic processes, unlike mechanical processes, are not reversible
In the human being, the temporal element is a reality, while in a machine the temporal element is only a function of space
Real time is not a fourth dimension as it is in Einstein's continuum
The world of time is actually the world of the plane of time; it is two dimensional
Its analogue in projective geometry is the boundary plane of three-dimensional space
This plays into what is called beholding in the imaginative world
Color perspective as another analogue of the imaginative world
Two dimensions become real in the imaginative world, one dimension in the inspired world
The intuitive world is pointlike
This cannot be referred back to Euclidean space, however
Dornach, December 29, 1922
159
4
Mathematics as a product of the human spirit
It is difficult to use mathematics to apprehend reality
The transition from a sphere to a projective plane
Concrete tasks for mathematicians: apprehending reality in mathematical terms--for example, explaining tactile and visual space in terms of differential equations, which must be integrated according to LaGrange's method
The variables for tactile space are positive, for visual space, they are negative
The difference of the integrals is nearly zero
Additional calculations yield equations for acoustics
We must learn to restrict our calculations to the domain of concrete reality
NOTES • PART I
163
23
NOTES • PART II
186
35
Bibliography
221
12
Glossary
233