# Mathematics of the etheric

Just as solid matter and its behaviour is described with a certain type of mathematics and laws, different forces and worlds are bound by different laws and require a different language [see 1921-06-GA205 references below]. This is because the language chosen by science for describing the mineral aspects of physical matter is very much fitted to the characteristics of what is described. However, the nature of other dimensions of nature is quite different, and the language or mathematical representation of mineral science breaks down, or is just not fitted to do this.

The etheric and its formative forces require such different kind of mathematics. In mineral matter centric forces are at work, causing solid substance, gravity and weight. The formative forces in the etheric are characterized by different properties of expansion and contraction and are peripheral (between the point and the plane) rather than centric..

Contemporary physics, as foundation for 'mineral science', is based on Euclidean geometrical thinking of space measured in finite and rigid lengths, with areas and volumes based on the measurement of length, with laws of parallelism and the right angle. The implicit underlying assumption being that this applies from the smallest to the largest dimensions. And as any frame of reference, it limits what we can see or describe in the frame. In this case the thinking of space is very much earthly, meaning related to our human sensory experience of vision and confirmation by tactile experience.

From the study of perspective in the middle ages arose a new kind of geometry that includes not just the finite forms but also the infinite space: vanishing lines and points of perspective. In that new geometry the infinitely distant is treated realistically, a bold step in thought.

Reasoning with this other geometry, complementary to the long-established Euclidean geometry, has two implications:

- focus is no longer on rigid fixed forms such as square or circle, but on mobile types of form that change depending on perspective or geometric transformation. Hence a conic section can result in a circle, ellipse, parabola, hyperbola. It is a foundational concept for qualitative spatial thinking about the metamorphoses of living form.
- principle of duality or polarity: projective geometry recognizes as the deepest law of spatial structure an underlying polarity which can be called, in simple and imaginative language, the polarity of expansion and contraction. Expansion and contraction of a sphere leads to two limits of a point or a plane. In other words: three-dimensional space can equally well be formed from the plane inward as from the point onward. These two polar opposite aspects represent the essence of spatial structure. Its implication is that whatever geometrical form or law is conceived, there will always be a sister form or law in which roles of point and plane are interchanged.

Now if these points are taken together, we start to think of universal space as a) not just pointwise but also planar (perspective outside in instead inside out) and b) with a balanced relation between contractive and expansive, or centric and peripheral qualities.

## Illustrations

## Lecture coverage and references

**Synthetic or 'projective geometry**' was developed around 1820-1840 by French mathematicians Jean-Victor Poncelet, Joseph Gergonne and Michel Chasles and independently by the Swiss Jakob Steiner (a pupil of Pestalozzi).

Later, around 1871 the concept of **path curves** or W-curves was developed by the friends and collegues Felix Klein (1849-1925) and Sophus Lie (1842-1899).

Rudolf Steiner already mentioned non-Euclidean geometry for study of the etheric and as a way out of the mineral scientific worldview in 1910.

The term of '**counterspace**' and the idea of 'peripheral forces' (re centripetal & centrifugal) and 'suction', was introduced by Rudolf Steiner in the lectures of 1920-03-11-GA321 (where he really introduces this for the first time), re-iterating it in and calling it 'counter-space' in 1920-05-02-GA201 and elaborating it further in the 1921-01-GA323 lectures.

See also: Four kinds of lawfullness in 1921-06-24-GA205 and 1921-06-24-GA205

Projective geometry and path curves were taken up further by mathematicians George Adams Kaufmann (1894-1963) and Louis Locher-Ernst (1906-1962), and continued by Lawrence Edwards and Nick C. Thomas.

## Discussion

### Counterspace terminology

Note the concept of 'counter-earth', referring to counterspace (see mathematical framework), already appears in Pythagoras (Antichthon), and also Aristotle described it. According to Aristotle the function of the Counter-Earth was to explain "eclipses of the moon and their frequency".

See also: Threefold Sun

## Related pages

## References and further reading

- George Adams: 'Physical and ethereal spaces'
- Olive Whicher: 'Projective geometry'
- Nick Thomas
- Science between space and counterspace
- Space and counterspace

- Oliver Conradt: 'Mathematical Physics in Space and Counterspace'
- For another example of contemporary work on projective geometry, see eg the work of Charles G. Gunn (profile).