The Physics of Higher Dimensions
Rough Draft - Introduction
This is one of a series of sites on the thorny questions that physicists did not answer last century. Perhaps it will provide an insight into those challenges that you will have the opportunity to address during your career as well as your personal life. On this site we shall explore the implications of the existence of more than Three Dimensions. An attempt will be made to examine the mathematical, scientific, technological, philosophical and other human implications of 4 or more spatial and/or physical dimensions.
I will take advantage of the interactive and multimedia properties of digital technology to provide an intuitive understanding of the physics of higher dimensions. There are many mysteries of science that can be easily explained with the use of higher dimensions. Among these are EMF and Gravity, Quantum Mechanics, Gravity, Dark Matter, Intrinsic Spin, The Hierarchy Problem, Particles vs. Waves, GUT's, TOE's, Parapsychology, UFO's, Angels, Ghosts, and Missing Socks (the latter will be left as an exercise for the student). An explanation of why these are Mysteries is provided here.
In this section I will provide the clues, you work out the details. Scientific Facts and currently accepted Theory are presented in black text. Musings, What-If's and Speculation are presented in magenta. Do not lose sight of what is Theory and what is Speculation.
|Table Of Contents|
|1.1.1||Open 3D Viewer Window|
|1.2.1||Open 4D Viewer Window|
|1.3||Other 4D Viewing Options|
|1.3.1||W by X perspectiveless projection along Y axis|
|1.3.2||3D Cross Sections of 4D Object|
|1.3.3||Other 4D Objects|
Our 'reality' is a world of 3 Space Dimensions and 1 Time Dimension. We believe this because we can see 3 dimensions - up/down, left/right, and front/back. We can also believe that there is 1 time dimension - future/past. This is known as 1+3 dimensionality - 1 Time dimension and 3 Space dimensions.
But could there be different dimensionality than what we perceive? Is it possible that there are 4 spatial dimensions? Or 2 time dimensions? Or more? These alternatives can be referred to as HyperDimensionality. We can simulate these alternatives with the computer in order to get an intuitive grasp of these hyperdimensional relationships.
On these pages we shall explore our 1+3D 'Reality', and a few of these alternate HyperDimensional worlds.
First let us practice with our common 3D world. Lets ignore Time and stick with 3D rather than 1+3D. I have provided a simple CAD-like viewer so that you can examine a 3D cube. Click here to open another window with the 3D viewer JAVA Applet. NOTE: You must have JAVA 2 installed to use the viewer. If you do not have the current JAVA installed, the viewer will tell you, and take you to the SUN JAVA site to download a copy. You must have the JAVA 2 (JRE 1.3 +) to proceed with this interaction.
3D Cube - A cube is 3 pairs of 2D surfaces, one pair for each dimension, attached at their adjacent 1D edges. So there are 6 2D surfaces. Each surface has 2 pairs of 1D edges, one pair for each dimension. There are 4 shared 1D edges for a total of 6*4 / 2 = 12 1D edges.
The 3D Viewer shows 3 views of a cube. TOP, FRONT and SIDE. Note that there are 3 labeled axes within the cube - X, Y, & Z. Notice the orientations of each of the axes in relation to the view. The FRONT view orientation is as you would expect - i.e. the X axis is horizontal, while the Y axis is vertical. The Z axis extends out of the window towards the viewer. The TOP view is indeed looking down on the cube from the +Y axis, while the SIDE view is looking along the X axis from the right.
|3D: Screen X||disabled|
|3D: Screen Y||disabled|
|3D: Screen Ctr||disabled|
SPIN Button - Click on the SPIN button in the bottom right of the viewer. Notice that the cube is spinning about the screen's Horizontal X axis (which is coincidentally the same as the cube's X axis). Look closely until your are sure that these are 3 views of the same cube from 3 different positions. In each view, the X axis is stationary, and the cube spins about it.
CENTER Button - Note that the axes spin with the cube. These are Object axes, not view (or world) axes. Note also that the Y axis appears to be rotating into the Z axis' space. This will be important later. Click on SPIN again, and the spinning will stop. Click on CENTER, and the cube will be centered back in its original position. In the FRONT window this is with X oriented along the window's horizontal axis, Y along the window's vertical axis, and Z coming out of the window towards you. This is a Right-handed Coordinate System (RHCS).
Screen X Slider - The first slider (the top on the left) controls the 3D object's rotation about the screen's Horizontal (or 'X') axis. If you slide the 'thumb' from the left to the right, the FRONT view will dip down. The SIDE view will twist anti-clockwise, and the TOP view will dip towards you. Click on CENTER to reorient the cube.
Screen Y Slider - Click on SPIN a third time, and the cube will spin about the next slider in sequence - the Vertical Y axis - in all 3 views. Here the Y axis is stationary while the Z axis rotates into the X axis. Click SPIN once again to halt the spinning. Click on CENTER to reorient the cube.
The second slider controls the 3D object's rotation about the screen's 'Y' ( or vertical) axis. If you slide the thumb of this slider from left to right, you will see the object rotate from left to right in the FRONT view. The TOP and SIDE views will rotate appropriately. Click on CENTER to reorient the cube.
Screen Ctr Slider - Click on SPIN a fifth time to activate spinning about the Z axis, or the screen's center. Here Z is stationary, and the X axis rotates into the Y axis' space. Click on SPIN once more to halt. If you clicked SPIN again, it would repeat the 3 slider sequence. Click on CENTER to reorient the cube.
The third slider controls the 3D object's twist about the screen's center. If you slide the thumb from the left to the right, the object will rotate counterclockwise. (The bottom edge of the FRONT view moves with the slider from left to right & vice-versa.) Click on CENTER to reorient the cube.
Perspective Slider - The bottom slider controls perspective - i.e. the relationship between an object's distance and its apparent size. Minimum perspective is at the left, maximum to the right.
As you slide the thumb to the right, the perspective gets more extreme. It is as if you move your eye up near the object - the closest square gets huge compared to the further square. The absolute size (or scale) of the projection is automatically adjusted to prevent overflow (very large senseless numbers).
As you slide the thumb to the left, the perspective is reduced. The apparent sizes of the back and front squares will look the same. It is as if you leaned way back away from the object (but increased the magnification so it stayed the same size). The perspective slider thus allows you to adjust the perceived depth of the object. A value of around 7 seems to work best for me.
Visual Cues - You can manipulate the other sliders while SPIN is active. If you get lost (can no longer 'see' the perspective) remember that the longer edges are closest to you, while the shortest edges are furthest from you. You can also press the CENTER button at any time to reorient the cube.
Play 3D Games
Game 1 - The 3rd button is a game. Press the SCRAMBLE Button to randomly orient the cube. Use the sliders to return it to the center position. (Note that the scrambled slider values have no meaning - in order to confuse you). Try the SCRAMBLE and restore game a few times until you feel comfortable with the 3D Viewer tool.
Game 2 - Place the green square inside the red square in the FRONT view. This will seem less trivial when we get to 4D. You can close the 3D viewer window when you are done.
In this section we shall explore a 1+4D model. To simplify the discussion, we shall ignore the time dimension, and stick to 4 space dimensions or 4D.
I find it easiest to visualize a 4D object by extrapolating the behavior of a 3D to 2D projection into a 4D to 3D projection. For example, a 3D cube is built of 6 2D squares attached at their 1D edges. To extrapolate into a higher dimension, imagine a 4D hypercube built of 8 3D cubes attached at their 2D faces. You can see that there may be mathematical patterns that can be extracted from these extrapolations. But better yet, lets take a look.
The 4D interactive visualization tool is an extension of the simple 3D viewer discussed above. Click here to start the 4D Viewer in another window.
4D Hypercube - A hypercube is 4 pairs of 3D cubes, one pair for each dimension, attached at their common 2D surfaces (sound familiar?). So there are 8 3D cubes of 6 faces each, each face shared between 2 cubes, this yields 8 * 6 / 2= 24 surfaces - or 12 pair, three pair of 2D surfaces for each dimension, or one pair for each of the 3 dimensions along each of the 4D axes.
Extrapolating, it would seem that while an infinite 2D plane can bisect an infinite 3D universe, it would require an infinite 3D object to bisect an infinite 4D universe.
The 4D object shown here is a unit hypercube. It is a 4D object that is 1 x 1 x 1 x 1 (while a unit 3D cube is 1 x 1 x 1). Each of the identical cubes of which the hypercube is formed is a 1x1x1 unit cube.
4D to 3D - The Interactive Visualization that follows provides a projection of this 4D hypercube into a window. This is done by projecting a 4D hypercube into a 3D space thus creating a 3D object, in exactly the same way that 3D is projected into 2D. The resultant 3D object is then projected onto your 2D window with the 3D viewer. This is mathematically equivalent to the prior 3D visualizer. If you lose it (i.e. - can no longer see the image in proper perspective), remember that the bigger edge is closest to you (and the shortest edges are further into the screen from you).
4D Viewer - There are 4 windows. The lower left window is a FRONT view of the hypercube. The lower right window is a view of the object from the right SIDE, while the upper left window is a view of the object from the TOP. These views are identical to the 3D viewer. The upper right view is a view of the object in the 4th dimension (more later).
|3D: Screen X||4D: Object W<=>X|
|3D: Screen Y||4D: Object W<=>Y|
|3D: Screen Ctr||4D: Object W<=>Z|
|3D: Perspective||4D: Perspective|
The left column is the 3D viewer, with which you are now facile thanks to your success with the 3D Games, above. The right column, which is now enabled, controls the 4D rotations and projection. The controls on the right are used to 'build' a 3D object from the projection of the 4D hypercube into 3 Space.
Remember to click on the CENTER button to reset the view to the nominal CENTER position. The 3D Horizontal, Vertical and Center Sliders on the left are as described above. So let us now focus on the 4D sliders on the right.
W Axis - Now that you are familiar with the conventional 3D viewer, we can get on to the fun stuff - the 4D controls - the sliders on the right. It is time to introduce the 'W' axis - the fourth spatial dimension. In the following discussion, I shall use 'W' to describe the new 4th space axis. The 4D object is colored to show the new dimension. The positive direction along the W axis is indicated by green edges, while the "-W" direction is shown with red edges. So, the green cube lies on the +W axis, while the red cube is in the -W direction.
Hypercube - The 4D figure shown here is a unit Hypercube. It is a 4D object that is 1 x 1 x 1 x 1 (a 3D unit cube is 1 x 1 x 1). A Hypercube consists of 8 identical cubes, each sharing its 6 faces with the 6 adjacent cubes (in the same way a 3D cube shares each of its 4 edges with the adjacent 4 squares to form a cube of 6 faces). Each of the identical cubes appear to be different sizes and shapes due to perspective. They really are identical. The 4D viewer lets you discover this truth as you explore the 4D object.
If you rotate the 3D object about Y (use the Screen Y slider) to 30 or 40 degrees, you will see a red cube inside a green cube. The red cube is the same size as the green cube. Don't believe it? Perspective makes the red cube (which is further away along the W axis) appear smaller than the green cube (which is closer than the red cube) - just like the red & green squares in the 3D viewer.
4D Rotations - Consider rotations. In 3D we think of rotation about an axis. However, this is equivalent to rotating one axis into another - i.e. rather than rotating about the Y axis, rotate the X axis into the Z axis. In this manner we can extrapolate rotations into 4D. The X axis is rotated into the Z axis (about the Y,W axes pair). This is convenient mathematically, since we can use graphical 3D rotational matrix transforms to describe these rotation(s).
In 3D space a plane is rotated about an axis, or a plane is rotated about all the remaining axes: 3D - 2D = 1D. Now in 4D, a plane is rotated about the two remaining axes, which describe a 2D plane: 4D - 2D = 2D.
Normal - If I choose to characterize a 4D normal to a plane as another 2D plane composed of the complementary 2 axes (e.g. the XY plane is rotated about its normal, the ZW plane) then the system has some useful symmetries. Note that the planes are orthogonal to each other in all 4 axes. There are no axes in common. This condition cannot occur in 3 Space.
Cross Product - Having defined the normal as the complementary plane, we can define the cross product of two 4D vectors in a 4D plane as this same complementary 4D normal plane.
Symmetry - The normal to the normal XY plane, is the original (XY) plane. Since this completes the dimensionality of our 4D space (2D + 2D = 4D), the rotations are complementary - i.e. does the XY plane rotate realtive to the ZW plane, or does the ZW plane rotate about tthe XY plane? Consequenty, there are only 3 possible rotations in our 4D space: XY/ZW, XZ/YW, and XW/YZ. Same rotational degrees of freedom available in 3D space. Symmetry implies conservation. What conservation does this rotational symmetry imply about 4 Space?
You will note, that the sliders on the right (again we shall ignore the perspective slider) are labeled 4D with an axis pair (be sure to click on CENTER for each of the following 3 paragraphs).
4D W<=>Z - The first slider (top right) rotates the object's W axis into the Z axis. As the slider's thumb is moved from the left to the right, the Z axis data is transformed into W axis data. Rotate about XY plane.
Observe the Red,Green,White (XYZ) axes in the SIDE view. You can see the W axis grow, and the Z axis shrink (and vice-versa). The SIDE view appears to be rotating about the Y axis in the SIDE projection. But as implied above, the rotation is actually about the X,Y pair.
Slowly drag the "W<=>Y" thumb from the 0 position to the 90 position, and note that the SIDE view axes show the W axis growing to replace the Z axis which is shrinking. At exactly 90 degrees, the Z axis has been replaced with the W axis.
Likewise the TOP view shows an apparent rotation of W into Z, but about the X axis (really about X,Y pair).
Ouroborus Effect - In the FRONT view, the W&Z axes are colinear and perpendicular to the view plane, projecting into and out of the screen, respectively. The Ouroborus Effect becomes clear if you rotate the FRONT view by 30 degrees (move the 3D: Screen Y thumb to the right near 30). As you rotate the W into Z with the top right slider, the object will appear to "eat its own tail" as the 4D portion of the object moves into our 3D space and out again. You can see the red cube move out of the green cube, then expand to consume the green cube. The green cube then expands to eat the red cube again. The motion through the W dimension (along the W axis) appears as expanding and shrinking red and green cubes.
You can explore each of the rotations with the 2nd (rotate W into Y) and 3rd (rotate W into X) sliders to display effects similar to those described above.
Play 4D Games
The SCRAMBLE button randomly orients the hypercube in 4 Space, and scrambles the sliders. The sliders work as before, just the value (at the right of the slider) and the slider position no longer correspond to the rotation of the object. Notice that every time you click on SCRAMBLE, the hypercube is reoriented. The CENTER button takes it back to the reset position.
The SPIN button will automatically spin each of the 6 rotation sliders in turn. Click on SPIN the first time to automatically slide the 1st (3D X Axis) slider. Click on it again to disable the spin. Click on it a third time to select the next slider (3D Y Axis), and a fourth time to cancel. Clink a fifth time for the next (3D Z Axis) slider, and a sixth time to cancel. Click on the SPIN button a seventh time to select the bottom right (4D Object W<>X), and an 8th time to cancel. Click 9 selects "4D Object W<>Y", 10 cancels. Click 11 selects the top right "4D Object W<>Z" slider, and click 12 cancels. Click 13 starts over again with "3D X".
Game 1 - Notice the position of the hypercubes when the CENTER button is pressed. Now press the SCRAMBLE button. Can you get the 4D object back into the CENTER position using just the 6 sliders?
Game 2 - Click on CENTER, then rotate the hypercube -30 degrees about the 3D Screen Y axis. Notice that the red cube is inside of the green cube. Can you place the green cube inside the red cube? Hint - it is a 180 degree 4D rotation about the W and one other axis.
Game 3 - Click on SCRAMBLE. Using the 6 sliders, get the red cube inside the green cube.
Game 4 - Try each of the above with a different axis spinning via the SPIN button.
W x X Projection Along Y
Shown here is an animated projection of 4D objects into 3 Space (and onto your 2D screen). These should be self descriptive. Click here for the animation provided by the University of Minnesota's Geometry Center. This animation will provide some unique insights into a spatial 4D.
Cross Sections (Notes for development)
The Cross Section feature slices a 3D object, rendering a 2D slice. By adjusting the position of the slice along the 'slice' axis, the shape of the 2D object will change. Rotating the 3D object across the slice via the 3D sliders will also change the shape of the 2D object. Lets extrapolate into 4D...
The Cross Section will also slice a 4D object, yielding a 3D slice. As with the 3D simulation, sliding the slicer along the 'slice' axis will change the shape of the 3D object. Similarly, rotating the 4D object across the slicer via the 4D sliders will also change the shape of the 3D object.
Now lets treat our 'slice' (W) axis as a time (T) axis. Sliding the slicer along this T axis works as above with the W axis. As the slicer moves forward and backward in time, the shape of the object 'morphs'. Click on the ANIMATE button to watch the slicer progress along the Time axis, morphing the 3D object. Click on the SPIN button to make it interesting.
In this section we shall explore alternate 4D metrics. For example, we can explore the implications of one compactified (rolled up) dimension. We will also explore 2 additional dimensions, where the added 2D metrics can be expressed not only as cylindrical, but as hyperbolic, spherical, elliptical, as well as infinite.
A work in progress ...
Additional 4D Viewer Features:
4) 3D cross sections. (Get old clipping code - modify to clip in 4D)
5) WxX projection along Y axis.( use orthogonal projection)(link to UMN animations)
6) Interpret W as Time (Create animations of 3D sections along W axis)
7) Use alternate metrics for W,
e.g. - Cylindrical, Hyperbolic, Spherical, Elliptical, as well as Infinite.
8) Other 4D objects.(link to polytope)
9) Use Dithering for probability.
10) Use shading for the W axis.
11) Hook 'amplitude' to the W axis.
12) Try 4D & 5D rotations (intrinsic spin interpretation)
13) Add Time axis (1+4D & 1+5D)
14) Link to Compton Pool Sim (ComPoSim)
15) Link to Relativistic Pool (RePoSim)
16) Link to Magnetic Relativistic Pool Sim