The Fourth Dimension

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ISBN: 9780880104722 | 0880104724
Cover: Paperback
Copyright: 7/1/2001

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The point, line, plane, and solid objects represent the first three dimensions, but a kind of reversal of space is involved in the ascent to a fourth dimension. Steiner leads us to this new perspective with words, diagrams, analogies, and examples of many kinds. In doing so, he continues his life-long project of demonstrating that our objective, everyday thinking is the lowest rung of a ladder that reaches up to literally infinite heights. Translated into English [from GA 324a] for the first time, these talks and the other selections on many mathematical topics bring us to tantalizing new horizons of awareness. They discuss: -- The relationship between geometric studies and developing direct perception of spiritual realities-- Constructing a fourth-dimensional hypercube-- Creating dimensions through movement-- Six dimensions of the self-aware human being-- The relationship of the Trinity and the angelic hierarchies to physical space

PART I LECTURES ON THE FOURTH DIMENSION

About this Edition

xiii

Introduction

xvii

David Booth

First Lecture, Berlin, March 24, 1905

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(1)

Plants, animals, and human beings as four-, five-, and six-dimensional beings, respectively

Basel, October 1, 1911

(1)

Light possesses inwardness as the fourth dimension

Munich, November 25, 1912

(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

(1)

The occult significance of the Golden Section

Berlin, November 27, 1913

(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

(1)

Written answers to questions on mathematics

Stuttgart, March 7, 1920

(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

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The Fourth Dimension

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