The Geometrical Doctrines of Physical Space, the Flatland of Abbott

7/25/2019 The Geometrical Doctrines of Physical Space, the Flatland of Abbott

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The geometrical doctrines of physical space, the Flatland of Abbott .

From my book www.mpantes.gr

The flatland

The A. Square

The doctrine of Riemann

The doctrine of Poincare

The Flatland, the A. Square

One of the more effective methods for imagining the fourth dimension is the

method of analogy. That is, in trying to imagine how 4-D objects might appear to us , it

is a great help to consider the analogous efforts of a 2-D being to imagine how 3-D

objects might appear to him. To understand the physical space problem, we will move

fantastically in a world of two-dimensional beings living on a positive curvature surface,

a huge sphere, which, as is known from the analysis of the elliptical geometry,

behaves like the Euclidean plane for small areas of our daily activity.

The country of two-dimensional is called Flatland, its inhabitants are tiny flat

shapes-beings, eg Mr. A. Square, a square of side a, likewise B. equilateral Triangle,

etc. that living in this huge spheres surface believe that areliving in a Euclidean plane

as they are plane creatures. Thus they are like us, except that we have the supervision

of the three dimensions! What it means for A. square the third dimension? What it

means for us the fourth.Through the two-dimensional we can imagine non-Euclidean

spaces although Kant argued that this is impossible, the space and time, say, are
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properties of human consciousness and not properties of the world. But the two-

dimensional image of, helps and puts on the stage, the revolution of Riemann.

The A. Square first appeared in the book Flatland, written by Edwin A.Abbott

around 1884. It is not clear if Abbott was actually the originator of this method of

developing our intuition of the fourth dimension. In his book, Rudolf Rucker,

"Geometry, relativity and the fourth dimension" describes hilarious images from the

lives of residents of Flatland, and this makes us understand the engagement of the

fourth dimension in our world. Here is Mr. A. Square who can move up/down or

left/right or in any combination of these two types of motion , by he can never move

out of the plane. He does not Know but only these two directions on the plane. He

is completely oblivious of the existence of any dimensions other than the two, and

one day a miracle happened in his life. This was the contact of A. Square with the

Third Dimension: the A- sphere, an intelligent being of three-dimensional space,

approached the level of the country of Flatland. When the A-sphere first came into

contact with the 2-D section, A-Square saw a point and as A. Sphere continued his

motion the point grew into a small circle which became larger and then smaller and

finally shrank back to a point, which disappeared. Figure (1)

A-squares interpretation of this strange apparition was he must be no circle

at all, but some extremely clever juggler. And what should we say if we heard a spectral

voice proclaim, I am A-hypersphere I will teach you the fourth dimension being a

point which slowly into a sphere which then shrank back to a point and finally was out

of existence!


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The difference between the two experiences, says Rucker, is that we can easily

see how to stack the circles up in the third dimension so as to produce a sphere, but it

is not at all clear how we are to stack the spheres up in the fourth dimensions as to

produce a hypersphere (a sphere in the four dimensions).

Rucker describes this contact of A. Square with the sphere, as the beginning of

an adventure and risks for him. The sphere came back and insisted speaking for the

third dimension, but A. Square remained unconvinced, so A. Sphere did some more

tricks. First he removed an object from a sealed chest in A.Squares room without

opening the chest and without breaking any of its walls. How was this possible? A

chest in Flatland is just a closed 2-D figure, such as a rectangle. But we can reach in

from the third dimension without breaking through the trunks walls. The analogy is

that a 4-D creature should be able to remove the yolk from an egg without breaking

the shell! Everything on Earth is opened to a D-spectator even the inside of our heart. A

lot of people used to think so at the time of Spiritualist movement around 1900. The

idea was that spirits were 4-D beings who could appear and disappear at any point see

everything and so on.

The only way in which A. Sphere could finally convince A. Square of the reality

of third dimension, was to actually lift him out of Flatland.

He explained that after such

a trip could return as his own reflection in the mirror, he would have another

orientation in the plane as in Figure 2. The analogous in us means that after a similar

trip in the fourth dimension we would return to our bowels outside and our

skin from the inside! All such terribles!


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Finally A. Square believed in the Third dimension and became a preacher. He was

persecuted by the priests, but with the intervention of the A.Sphere he was saved and

remained throughout his life a proponent of the idea of the third dimension.

The linking of the experiences and conclusions of the Flatlanders with our own experiences and

conclusions, will be apocalyptic.. Through the Abbotts Flatland we will follow our own path to explore

our physical space, making frequent comparisons.

Centuries ago this adventure of A.Square, the people of Flatland cultivated science and

geometry. Their Euclid, taught them the Euclidean doctrine (5th axiom) and interpreted the

world around them, considering it as flat and endless, but with a thorn in their mind: why

we apply the 5th

axiom? How do the lines will behave at infinity?

Nevertheless their geometry was the standard of scientific thinking of Flatland. Andwhen many years after, their technology has developed too far, the two-dimensional

intelligent beings decided to do a large-scale experiment in order to check their geometry.

Therefore conducted an enormous triangulation with light rays, in order to measure the

sum of the angles of the triangle formed.

In our history, Gauss appears to have been the first to undertake space explorations of

this sort, when he performed triangulations with light rays from one mountain top to

another. But his observations were too crude and executed over too small an area to detect

any trace of non-Euclideanism. Lobatchewski also suggested astronomical observations

conducted on the course of rays of starlight through interstellar space.

So the Flatlanders measured the angles of the triangle extended triangle in their putative

flat world, and at the same time brought the great crisis in the science of their society. Each

triangle which was counted, had a different angles sum and no 1800. This crisis could be

compared with our own confusion in the results of the Michelson-Morley experiment, lead

to deadlock all our ideas about space and time.

So the light triangulations was the trigger for the two-dimensional to start exploring

their physical space. They understood that the results of the light triangulation were not

Euclidean. And yet there was no way to find out if they were on a sphere or on a flat sphere

(the familiar plane where we installed the spherical geometry), which as we know are

isomorphic spaces. They had, then, to consider the space as an Euclidean plane and change

their view about the light path which until then was considered as Euclidean straight line, or


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maintain the straight orbit for the light deciding that are living in a sphere, recognizing this

mysterious third dimension.

So the proposals that have dominated for the interpretation of the findings of the light

triangulation, arrived at two considerations, in that of Riemann and to that of Poincare.

Riemann in a classical mathematical formulation in his article "assumptions underlying

geometry" drew the attention of two-dimensional to a distinction that seems so obvious as

established: the distinction between the unlimited straight line and infinitely extended. The

difference is easily understood: a circle is an infinite line that never ends, but has a finite

length. On the other hand, the Euclidean straight is also unlimited in the sense that never

ends, but it is of infinite length. This change of the concept of straightness, implies for

Riemann, a new way of measuring distances, the formula

2 =2 + 2 + 2

1 +2 + 2 + 2

2

which is the differential expression of the known type for the e-distance. In this type of

Riemanns line element, the interpretations of k created our two basic geometric

doctrines for the physical space of our (their) world.

Riemanns doctrine

The term k for Riemann is the curvature of the space, that is due to a

property of space. He talked about the third dimension and curved spaces, taught the

elliptical geometry of the sphere and of course the first fan was A square. But all of this

could not be certified, and the fundamental truths about space, Riemann says are

deduced from experience. So Riemanns views were ignored for many decades till

Einstein created the general theory of relativity. It is analogous to the view that we are

three-dimensional, and live in the curved surface of a hypersphere! Now we

understand the difficulty for the two-dimensional: the isomorphism established by the

stereographic projection it just means: the two-dimensional although understand that

their geometry is not Euclidean, they can not prove if they live on a sphere or a flat

sphere. (The Euclidean model of elliptic geometry). Lack of experience on the third

dimension conceals a direct observation. The two-dimensional experimenter can not

detect any curvature on the straight line, curvature of the kind the Riemann suggested


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that across the third dimension. Moving onto a geodesic of sphere, he says with

certainty that moves in Euclidean straight line, he does not "break" neither right nor

left. Similarly we can not find any of our curving line beyond the three known

dimensions, eg curvature at the time dimension, so we are in the same position as the

two-dimensional: the proposal of Riemann we have to change the geometry, the

space is not Euclidean, is curved at the third dimension and can no longer to measure

distances by Euclidean geometry" was not readily understood. But it is to Riemann

that they owe the insight that physics and geometry are inextricably mixed in the

problem if space.

The doctrine of Poincare

But another solution of the problem suggested by Poincare: the attitude of

the pure mathematician. For him the term k is not due to a property of space but in

physical conditions on the bodies of the space. He claimed, as in the geometric model

of a flat sphere, that space was irrelevant to the strange effect of light triangulation,

but the bodies and their behavior in the space, created the deviation from Euclidean

geometry. Space is not an entity such as the matter or the light beam. It is a

mathematical structure that enables us to describe the behavior of bodies in it, and the

question what is the correct geometry of space is meaningless as the question for the

correct measuring units. It is the meter or the yard? He considers undeniable that the

distances should be measured using the formula (1 ) as we have seen in previous

chapter, but his main difference is the interpretation of the term k. The term k alters

the familiar Euclidean distances in the infinity small, from place to place ( the x, y, z of

the formula), as to disfigures the bodies into space. It follows that by a mere variation

in physical conditions the same space would be considered non-Euclidean or Euclidean.

Obviously, by reason of this contradiction, space itself can have nothing to do with the

problem. The type of space which physicists are discussing reduces therefore to a

relational synthesis of physical results. Space itself remains amorphous. Really: If the

moving body is the measuring rod of the distances


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thus no system of axioms may claim that it is the true geometry. The principles of geometry are

not experimental facts, and in particular Euclids postulates cannot be proved by experiment

(Science and hypothesis)

So what seems clear is that direct experimental evidence in favor of one or the

other proposal doesnt exist for the Flatlanders. They lived in a sphere, but how to

prove the existence of curvature, since this was the mysterious third dimension? But

the deformable bodies could not be determined experimentally. Because for a such

proof should be a solid body in the world that would measure the deformation of the

other bodies, this being rigid. But where to find such a body since from deformation

not excluded anything? And if there was such a thing, one could say that this was

deformed rather than the bodies of their space. So the two proposals seemed

equivalent:

they could change the geometry holding physics unchanged (Riemanns view)

Or change the physics, keeping unchanged the geometry (Poincares view)

George Mpantes mathematics teacher

Books

www.mpantes.gr

D Abro.A. (1950) the evolution of scientific thought from Newton to EinsteinN Y .Dover

Rucker R.V.(1977) geometry ,relativity and the fourth dimensionN.Y Dover
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