TimeTraveller Order Page
The TimeTraveller is a Sacred Geometry offered by the One Earth Network as a tool for personal and planetary transformation. More about it here

On this page you will find illuminations of some basic concepts in higher dimensional geometry.

Getting a grip on what a fourth spacial dimension is ~ Different Projections of the Hypercube
Stereo Photographs and Binocular Vision

The Cosmic Christ and the Cross of Matter
~ Plato's - the Allegory of the Cave
Other Imagery
~ The Fifth Dimensional Time Traveller

 

Visualising four dimensional geometries does not come easily to a mind used to operating in 3 spacial dimensions. As far as mathematicians are concerned though there is no reason why there cannot be more than 3 spacial dimensions at right angles to each other. Location in 3 dimensional space can be defined by reference to 3 axis', as in - 'go forward 20 paces, right twenty paces, and then dig down 20 feet to find the treasure'. Location in four dimension would require an extra co-ordinate to define where a location is. Since we cannot function in a fourth spacial dimension Time can serve as an example of another necessary co-ordinate. For instance, 'but only dig down on the December solstice of 2012 or it will not be there'. :-)
We have neither the space nor time here to completely explain the theory and implications of four dimensional geometry, but we have made some introductory materials and collected some resources for anyone who wishes to learn more. On this disc there is the classic book 'Flatland - A Romance in many Dimensions' by Edwin Abbott (1838-1926). Click here to access the whole book as a text file, courtesy of the Gutenberg Project site. An even better classic book is Claude Bragdon's 'Primer of Higher Space' and you can find it and many other books about higher space at the One Earth Bookshop.

Here are a couple of animations that explain the structure of the hypercube...

The animation below shows the progression from point to line to square to cube to hypercube. Each step involves moving the previous figure away from itself into the next dimension. Each figure splits to become a red and a blue version of itself which then move apart but stay connected by white lines.The animation then shows the 8 cubes within the hypercube

This animation shows that when we draw a cube on paper at least some of the squares must be skewed. Then it shows the hexagonal shape where all the squares are equally skewed. The TimeTraveller projection is the 8 sided 4 dimensional equivalent of this which is why it is so strangely symmetrical.

 

Just as there are many different ways to draw a projection of a cube on paper, so there are many different ways to construct a projection of a hypercube in three dimensions, or two dimensions. Usually in textbooks or on the Web you will find one of these two projections ...
(colour roughly indicates the third dimension)

which is the equivalent of this

(not intuitively very cube like)
or
which is the equivalent of this

(not true perspective)

The first one of these is a small cube inside a larger cube, and the second one is two proper cubes with a flat red cube at the front and a flat blue one at the back.in both of these projections some of the eight cubes are much more distorted than the others. The TimeTraveller projection is far more symmetrical and all the cubes are identically distorted.

which is the equivalent of this

This projection is rumoured to have been mentioned in a book written circa 1930, but it was independently discovered by Firebird in the early 1980s.

This projection could be called the toroidal-hypercube because of its noticeably donut-like toroidal shape. ( Note the apparent 'hole' in the centre of this projection and its equivalent cube )

The first example here with the small cube inside the larger cube is perhaps the easiest to understand. Below is an animation which represents rotating the actual hypercube that is casting the 3 dimensional shadow.


This is a 3D stereo pair of photographs.
To see in 3D cross your eyes so that you begin to see 4 images
then merge the 2 middle images into 1.
This may sound hard yet we do something similar all the time.
The way that we unconsciously create a 3 dimensional perceptual world from merging the left and right eye's two dimensional (and upside down) views into one is truly astonishing to contemplate. Studying binocular vision can teach us much about dimensions, and deconstructing our assumption of 3 dimensional visual space is a very powerful step towards breaking the hypnosis of the senses!

 


This is the same pair of images swapped left and right so that
to see it in 3D you have to 'look through' the screen
and again merge the centre images into 1.
Some people find this easier and in this instance you will notice that the 3 dimensionalisation is more exaggerated.

To take stereo photographs like this it is only necessary to move the camera from one eye to the other whilst holding your head still, thereby recording both the viewpoints that can be reassembled by our brains into the 3 dimensional perspective again. With some practice it is possible to learn to draw geometric shapes and even freeform drawings in 3D.

TimeTraveller Order Page

This is an animated 'magic eye' type wallpaper - Cross your eyes very slightly to see in 3D
Try looking at the third or fourth row in to make it easier.

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Used with permission of www.divinedimensions.com


This graphic above is called the HyperMan. Combining elements of Salvador Dali's Cosmic Christ painting with Leonardo da Vinci's VetruvianMan with the image on the shroud of Turin, it represents the soul navigating in higher dimensional space. As you can see from the 3 pictures below Dali was obviously fascinated with the idea of the Christ's connection to other dimensions.
The form of the hypercube on the left below is the unfolded hypercube and is the equivalent of unfolding a cube to create the cross on the left here. The cube and the cross have both been used throughout history as a symbol for matter, existence, and space - within which divinity is born, dies, and is resurrected.


~ Paintings thanks to Salvador Dali ~

The Tesseract represents the cross of spacetime in which Spirit is inprismed to make visible the rainbow of existence.

To see more Sacred Geometrical Artifacts including this tiled box of the Hyperman at our
One Earth CafePress Shop


 

It is truly astounding that Plato should have written this most powerful metaphor for our situation nearly 2500 years ago, and without the benefit of having watched countless movies and TV programs! Plato says the whole of most peoples worlds is a projections on the 'wall of the cave' . Only those with what he calls 'philosophical training' can learn to distinguish the true world and the light of goodness that casts the shadows of appearances.
Its also remarkable how applicable this allegory is to our situation in trying to imagine a 4th dimension, especially considering that Plato was not specifically referring to higher dimensions but only the ability to distinguish shadows from the reality that causes them.

Here is the complete dialogue written by Plato
for his character Socrates and his student Glaucon

BOOK VII of 'the Republic'

 Socrates: And now I will describe in a figure the enlightenment or unenlightenment of our nature - Imagine human beings living in an underground den which is open towards the light; they have been there from childhood, having their necks and legs chained, and can only see into the den.  At a distance there is a fire, and between the fire and the prisoners a raised way, and a low wall is built along the way, like the screen over which marionette players show their puppets.  Behind the wall appear moving figures, who hold in their hands various works of art, and among them images of men and animals, wood and stone, and some of the passers-by are talking and others silent. 

Glaucon: 'A strange parable,' he said, 'and strange captives.' 

Socrates: They are ourselves, I replied; and they see only the shadows of the images which the fire throws on the wall of the den; to these they give names, and if we add an echo which returns from the wall, the voices of the passengers will seem to proceed from the shadows.  Suppose now that you suddenly turn them round and make them look with pain and grief to themselves at the real images; will they believe them to be real?  Will not their eyes be dazzled, and will they not try to get away from the light to something which they are able to behold without blinking?  And suppose further, that they are dragged up a steep and rugged ascent into the presence of the sun himself, will not their sight be darkened with the excess of light?  Some time will pass before they get the habit of perceiving at all; and at first they will be able to perceive only shadows and reflections in the water; then they will recognize the moon and the stars, and will at length behold the sun in his own proper place as he is.

Last of all they will conclude:- This is he who gives us the year and the seasons, and is the author of all that we see.  How will they rejoice in passing from darkness to light!  How worthless to them will seem the honours and glories of the den!  But now imagine further, that they descend into their old habitations;- in that underground dwelling they will not see as well as their fellows, and will not be able to compete with them in the measurement of the shadows on the wall; there will be many jokes about the man who went on a visit to the sun and lost his eyes, and if they find anybody trying to set free and enlighten one of their number, they will put him to death, if they can catch him.  Now the cave or den is the world of sight, the fire is the sun, the way upwards is the way to knowledge, and in the world of knowledge the idea of good is last seen and with difficulty, but when seen is inferred to be the author of good and right - parent of the lord of light in this world, and of truth and understanding in the other.

He who attains to the beatific vision is always going upwards; he is unwilling to descend into political assemblies and courts of law; for his eyes are apt to blink at the images or shadows of images which they behold in them - he cannot enter into the ideas of those who have never in their lives understood the relation of the shadow to the substance.  But blindness is of two kinds, and may be caused either by passing out of darkness into light or out of light into darkness, and a man of sense will distinguish between them, and will not laugh equally at both of them, but the blindness which arises from fullness of light he will deem blessed, and pity the other; or if he laugh at the puzzled soul looking at the sun, he will have more reason to laugh than the inhabitants of the den at those who descend from above. 

There is a further lesson taught by this parable of ours.  Some persons fancy that instruction is like giving eyes to the blind, but we say that the faculty of sight was always there, and that the soul only requires to be turned round towards the light.  And this is conversion; other virtues are almost like bodily habits, and may be acquired in the same manner, but intelligence has a diviner life, and is indestructible, turning either to good or evil according to the direction given.  Did you never observe how the mind of a clever rogue peers out of his eyes, and the more clearly he sees, the more evil he does?  Now if you take such an one, and cut away from him those leaden weights of pleasure and desire which bind his soul to earth, his intelligence will be turned round, and he will behold the truth as clearly as he now discerns his meaner ends.  And have we not decided that our rulers must neither be so uneducated as to have no fixed rule of life, nor so over-educated as to be unwilling to leave their paradise for the business of the world?  We must choose out therefore the natures who are most likely to ascend to the light and knowledge of the good; but we must not allow them to remain in the region of light; they must be forced down again among the captives in the den to partake of their labours and honours.  'Will they not think this a hardship?' 

You should remember that our purpose in framing the State was not that our citizens should do what they like, but that they should serve the State for the common good of all.  May we not fairly say to our philosopher - Friend, we do you no wrong; for in other States philosophy grows wild, and a wild plant owes nothing to the gardener, but you have been trained by us to be the rulers and kings of our hive, and therefore we must insist on your descending into the den.  You must, each of you, take your turn, and become able to use your eyes in the dark, and with a little practice you will see far better than those who quarrel about the shadows, whose knowledge is a dream only, whilst yours is a waking reality.  It may be that the saint or philosopher who is best fitted, may also be the least inclined to rule, but necessity is laid upon him, and he must no longer live in the heaven of ideas.  And this will be the salvation of the State. For those who rule must not be those who are desirous to rule; and, if you can offer to our citizens a better life than that of rulers generally is, there will be a chance that the rich, not only in this world's goods, but in virtue and wisdom, may bear rule.  And the only life which is better than the life of political ambition is that of philosophy, which is also the best preparation for the government of a State.

This humorous extension of the ideas in Plato's allegory is used with kind permission of its creator
Joshua Bach of www.contraculture.com

There is an interesting article here about Plato's Cave and the archetypally potent movie 'the Matrix'

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This graphic represents 4 axis of vibrational harmonics at right angles to each other inside a Hypersphere.
Any 3 of the axis can be seen as being at right angles to each other in 3 dimensional space.
If you move between them till you can see all 4 at right angles to each other at the same time
and you will have popped out of this universe :-)

This graphic is also available as Tshirts and more at the One Earth Cafe Press Shop

 

A hypersnowflake, or perhaps the field of a subatomic hyperparticle :-)

 

So the TimeTraveller is a symmetrical projection of the fourth dimensional cube into three dimensional space - what would be the equivalent projection of a fifth dimensional cube into three dimensional space ? We're glad you asked :-)

Here's a rough diagram of it. The vertexes are color coded for to give an indication of 3d depth. (Red at the front to yellow to green to cyan to the gray dot at the back.) Remember that once again that to be accurate all the lines have to be the same length.
One of our coming projects is to create this one as a sacred geometry item for the One Earth Network.

 

Click here to open an ebook of the complete text of Einstein's Special & General Theory of Relativity

There's an amazing animation of complex 4 dimensional shapes passing through 3 dimensional space here at www.dogfeathers.com/java/hyperstar.html

This site TetraSpace covers all the basics of understanding higher dimensional geometry very thoroughly

A pretty in depth exploration of 4 dimensional geometry
from basics to advanced visualisation is here

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