Transverse waves vibrate at right angles to their movement but light moves in every physical direction, so it must vibrate outside our space. If light vibrated in a physical direction, it couldn’t move that way, so space wouldn’t be isotropic (the same in every direction). In simple terms, after space gives light three movement dimensions, there are no free directions for a transverse wave to vibrate into, so a physical vibration can’t explain light waves at all.

But if our space is a surface then light could move on it as waves do on a lake, except in three dimensions not two. If light is a transverse wave vibrating outside our space, just as complex number theory describes it, then we are 3D “Flatlanders”.
In Abbot’s story, Flatlanders were beings who lived their lives on a surface that had only two dimensions (Abbott, 1884), so they could see a circle but had to imagine a sphere. Now suppose a point entity moved on their land but set values in a circle at right angles to it (Figure 3.8a). Flatlanders could then explain it as a vibration in an imaginary plane, just as we explain light in complex number theory. As the point moves, it would also have a frequency and a polarization plane in their space (Figure 3.8b), again as we have for light. They could then explain these moving points as imaginary vibrations, just as we do for electromagnetism (Figure 3.8c).

It follows that complex numbers could explain electromagnetism because light really does vibrate outside space, so:
“In quantum mechanics there really are complex numbers, and the wave function really is a complex-valued function of space-time.” (Lederman & Hill, 2004), p346.
Complex numbers describe light as rotating into a plane outside our space (Note 1), see Figure 3.9. Science calls this rotation imaginary because it doesn’t exist in our space, just as Flatlanders might call a rotation outside their space imaginary. But in their case, there really is a third dimension, so our case could be the same. If our space is a surface within a higher dimensional space, then light can vibrate into another plane as the equations say.
In the quantum model, our space is a surface inside a quantum network, so light can vibrate transversely. Quantum waves like light can’t leave that surface any more than waves on a sea can leave its surface, so if we are the same, we can’t leave our space. We are then three-dimensional Flatlanders, but what then vibrates when light does?
Note 1. Complex number theory describes a rotation into an imaginary plane. In normal multiplication, multiplying a number by two doubles it, e.g. 5 x 2 = 10. Multiplying by 4 adds it four times, e.g. 5 x 4 = 20. In complex multiplication, i is a 90° rotation into an imaginary plane, so times 2i is a 180° rotation that turns a number into its negative, e.g. 5 x 2i = -5. Times 4i is a 360° rotation that has no effect, so 5 x 4i = 5.