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Geometry, dimensions and hyperspaces
1. Ephesians 3:18 : From lowest depth to highest height
Ephesians 3:18 May be able to comprehend with all saints what is the breadth, and length, and depth, and height; [kjv]
εν αγαπη ερριζωμενοι και τεθεμελιωμενοι ινα εξισχυσητε καταλαβεσθαι συν πασιν τοις αγιοις τι το πλατος και μηκος και υψος και βαθος [gnt]
… latitudo … longitudo … sublimitas … profundum [v]
… breede… lengthe… hiynesse… depnesse… [wy]
Are there "
four" space dimensions? What is the difference between depth and height? Aristotle tried at length to verbalize the issues between abstract mathematics and observed reality in his works on
metaphysics.
The ancient Greek word
"πλατύς" ≈ "wide, flat" and is related to the English word
"flat".
2. Geometry, dimensions and hyperspaces
Isaiah 34:4 And all the host of heaven shall be dissolved, and the heavens shall be rolled together as a scroll: and all their host shall fall down, as the leaf falleth off from the vine, and as a falling fig from the fig tree. [kjv]
και ελιγησεται ο ουρανος ως βιβλιον και παντα τα αστρα πεσειται ως φυλλα εξ αμπελου και ως πιπτει φυλλα απο συκης [lxx]
Geometry is the study of space. Take a flat piece of paper that models a plane. To
roll it up, you need another
dimension. Another way to think about it is to "
fold" rather than
"roll".
There are many ways to represent dimensions. A dimension is a way of looking at something. Most people think of dimensions in geometry. Geometry is used here for a discussion on dimensions.
3. Isaiah 34:4
Isaiah 34:4 And all the host of heaven shall be dissolved, and the heavens shall be rolled together as a scroll: and all their host shall fall down, as the leaf falleth off from the vine, and as a falling fig from the fig tree. [kjv]
και ελιγησεται ο ουρανος ως βιβλιον και παντα τα αστρα πεσειται ως φυλλα εξ αμπελου και ως πιπτει φυλλα απο συκης [lxx]
4. Isaiah 34:4
KJV: And all the host of heaven shall be dissolved, and the heavens shall be rolled together as a scroll: and all their host shall fall down, as the leaf falleth off from the vine, and as a falling fig from the fig tree.
Hebrew: ונמקו כל צבא השמים ונגלו כספר השמים וכל צבאם יבול כנבל עלה מגפן וכנבלת מתאנה׃
Greek: και ελιγησεται ο ουρανος ως βιβλιον και παντα τα αστρα πεσειται ως φυλλα εξ αμπελου και ως πιπτει φυλλα απο συκης
5. Euclid
The Greek mathematician
Euclid of Alexandria (Greek mathematician) is famous for his works on geometry.
Fragments of Euclid's work still exist today such as a papyrus from about 2000 years ago found in the rubbish piles of Oxyrynchus in 1896-1987 by Grenfell and Hunt.
6. Euclid fragments
If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half. Translation of Euclid fragment by T. L. Heath.
From a mathematical and reality point of view, let us briefly look at the progression from point to line to plane to space.
7. Points
Mathematical points do not have any thickness (in the next higher dimension) so they do not exist in the real world - except in the mind.
In physics, an
electron can usually be treated as if it were a point particle. An electron has many of the properties of being a point object, but when one tries to measure position and/or momentum, the Heisenberg Uncertainty Principle limits the measurement process.
8. Lines
Any two distinct points in a plane determine a line in the plane. A line can depict one dimension, called 1D.
Two points, not the same point, determine a line.
A point on a line divides the line into two parts.
Mathematical lines do not have any thickness (in the next higher dimension) so they do not exist in the real world - except in the mind.
9. Planes
Three distinct points determine a plane. A plane can depict two dimensions, called 2D. A plane has no thickness (in the next higher dimension) so mathematical planes do not exist in the real world - except in the mind.
Two lines, not the same line, determine a plane.
A line in a plane divides the line into two parts.
10. Spaces
A space has no thickness (in the next higher dimension) so mathematical planes do not exist in the real world - except in the mind.
Four distinct points determine a space. A space can depict three dimensions, called 3D.
Two planes, not the same plane, determine a space.
A plane in a space divides the space into two parts.
Can the inductive reasoning be taken another step?
11. Dimension build
1 Point
2 Line
3 Plane
4 Space
Can the inductive reasoning of points to lines to planes to spaces be taken another step? Let us review the progression.
12. Human visualization
The purpose of visualization of concepts is often go get in intuitive feel/view for what is happening. The results and methods are then generalized to many more dimensions.
Humans have a unique ability to abstract and recognize patterns and make abstract inferences from those recognized patterns. Humans can easily visualize 2D or 3D in graphics but higher dimensions are harder to visualize.
Working in 2D or 3D can thus help one understand the method that then generalizes to higher dimensions. Since humans have trouble visualizing even four dimensions, the term hype-space will be used for 4D space. Unless otherwise specified, hyper-spaces will refer to 4 dimensions.
13. Hyperspace
Five or more distinct points determine a hyper-space. A hyper-space can depict four or more dimensions, called 4D (or more).
Two spaces, not the same space, determine a hyperspace
A space in a hyperspace divides the hyperspace into two parts.
Does this inductive reasoning have a basis in reality?
14. Hilbert spaces
The mathematician Hilbert generalized dimensions to a potentially infinite number of dimensions, called Hilbert spaces.
15. Dimension build
1 Point
2 Line
3 Plane
4 Space
16. Theological analogy to heaven and hell
According to many sources, heaven is somewhere up there and hell is somewhere down there.
The theological analogy is that the world we live in separates a hyperspace of four space dimensions into two parts.
Is there a physical reality to higher dimensions?
17. Einstein
Einstein's theories of special and general relativity imply that we live in a four dimensional hyperspace of three space dimensions and one time dimension in which both space and time are related.
18. String theory
The proponents of the physics of string theory claim that there are many higher dimensions above and beyond space and time.
19. Impossible things
Some apparently possible 2-D representations of 3-D worlds are impossible in the real physical world.
20. Escher waterfall
Water does not flow uphill by itself. One use of mathematics and models based on math is that it is easier not to be led astray by our own perceptions. Escher waterfall's are two dimensional depictions of three dimensional situations that cannot exist.
The Penrose steps are done in a manner similar to the Escher waterfall.
21. Salvidor Dali
Salvador Dali painting of the crucifixion, "
Hypercubic Body", from 1954, uses the a hypercube representation of the cross.
22. Straight lines
In traditional Euclidean geometry, lines are assumed to be straight.
Non-euclidean geometry allows lines to not be straight.
23. Negative numbers
Note, however, that until the development of "negative numbers", after Descartes and "Cartesian coordinates", height was above and depth below.
What is a "negative number"?
Negative numbers do not exist in reality. You should never have -3 cows in a field.
You have absolutely 3 cows in a field, so take the absolute value of a negative number (and see if it makes sense).
24. Going beyond the prefix para
The Greek prefix
"para" appears to mean
"beyond" and
not "parallel" which is the sense often used in English. The ancient Greek word
"παρά" ≈ "beyond", although some prefer the meaning of "
beside" because of the influence of Euclid's definition of a parallel line as a line
"beyond" another line in a plane such that the lines never touch.
From Euclid, a parallel line is a line just like another line in a plane, not that line, that is
"beyond" that line (from either direction) and never touches that line.
Some Greek words that make better sense using
"beyond" rather than "
parallel" include "
parable" and "
paradox".
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25. End of page
