We do consider scale as part of our reality.  We even consider it to be a continuum of reality, if not part of our 3-D model of space.  However, we experience objects as appearing differently at different scales.  Our bodies consist of cells and proteins, molecules and atoms.  We don’t see these at our scale, but at much smaller scales.  How can these different objects all occupy the same (3-D) space?

Four Dimensional Objects

Consider the tip of your finger.  Consider how you might specify the tip of your finger at different levels of scale:

  • The tip on our visual level
  • The tip on the cellular level
  • The tip on the protein level
  • The tip on the molecular level

What if you considered all these levels to specify the tip of your finger?  Do you feel like all these levels constitute the same point?  Or are they different points connected by some sort of line?  Which of these seems the correct interpretation, forgetting our three-dimensional bias?

Consider these points defining a line and not a point.  We should be able to determine the tip of our finger at each of these levels.  We could use a three-dimensional reference frame at our visual scale level and ‘slide’ it down to each level to measure the tip at that scale, changing the scale of the reference frame (see Figure 5).  Moving a three-dimensional frame of reference along a line not in this reference frame is essentially the definition of moving in another, in this case fourth, dimension.  Connecting the points of the tip of your finger at these different levels of scale would determine a line with a length not in any of our normal three dimensions – a line in a fourth scale dimension.

Finger

Figure 5:  Finger at different scales.

There is a well-known four-dimensional object, called a tesseract.  This is essentially a box extended into four dimensions.  Construction of a tesseract (see Figure 6: from Wikipedia http://en.wikipedia.org/wiki/Tesseract):

Construct Dimensions

Figure 6:  A diagram showing how to create a tesseract from a point.

0 – A point is a hypercube of dimension zero.

1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.

2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square.

3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.

4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).

Figure 7: Picture of a tesseract projected onto three dimensions (from Wikipedia).

Tesseract

(End of Wikipedia insert)

Now, consider both the construction of a tesseract and the tesseract projection in relation to the discussion of the tip of your finger.  Assume your finger is a three-dimensional object at our visual level and ‘move’ it in the direction of scale.  Wouldn’t a reasonable view of this be a four-dimensional finger that exists at different levels of scale?  And, if we smooshed (note the technical term) these levels onto a 2-dimensaional visual level – might a picture of your finger at these levels appear like a finger embedded in a finger?

Consider the tesseract (see Figure 7).  What would a three-dimensional cube existing at our visual level and, say, at the molecular level look like?  If we connect the edges of the cubes at these two levels, wouldn’t this appear like the cube-embedded-in-a-cube picture of a tesseract above?

There is an interesting feature of the tesseract: The ‘inside’ cube of the tesseract projection is actually another ‘lower’ surface of the 4-dimensional tesseract.  Everything that appears to be ‘inside’ of this lower cube is actually outside of the tesseract.  If the fourth dimension is scale, this leads to a surprising prediction – that we have a ‘lower’ scale 3-dimensional surface to our bodies and potentially all objects.  This is very different than our current conception – in which our bodies and all objects extend indefinitely into the realm of the smaller.  And if a particle much smaller than this surface whizzed by (say a neutrino), we would expect it to miss this surface.