Here I am running into terminology problems...again, but to say so, for me a 3 dimensional view of the Chladni plate pattern would incorporate the same contours to be on the bottom of the plate as well.

Terminology is very important so let's get agreement before looking at your question.

When I said "The chladni plate is in fact 3d because of the amplitude, think contour map" you might not have interpreted amplitude in the way I meant it.

Let's imagine you have an 8x4 sheet of hardboard and two 4ft trestles. Place the sheet on the trestles so that the trestles are widthways and about a 1/4 of the length from the ends (2ft). Now go to one end and press the edge down, the centre of the sheet will go up and the far end will go down, then move your edge up and the centre will go down, etc. This is a model of a rectangular plate in its 2,0 mode with the nodes at the trestles. If the dimensions of the sheet are x&y then z (perpendicular to x&y) is the amplitude of the vibration and is our 3rd dimension. Relative to the nodes up is +ve and down -ve. To think about the contours on the bottom of the sheet, when the top surface is domed the bottom is cupped, but they both move in the same direction.

Phase, when the ends go down, the centre goes up so although centre and ends are both antinodes they are antiphase whereas the ends are inphase.

It matters not to me if a 3 dimensional Lissajous figure is not not a proper world view...

OK, but I think you will agree that it is important to differentiate between objective observations and those influenced by the mis-perceptions of the brain.

All I'm interested in is the mathematical difference in phase shifting between the Lissajous and the Chladni.

Here I have a problem understanding what you mean.

So, using the terminology in the hardboard sheet example and similar examples, can you describe what you mean by "the phase shifting between lissajous and chladni".

Sorry Colin, I thought you were saying that the Chladni pattern is a 3 d pattern. Of course the plate and its vibrations are 3 dimensional.

In your hard board and trestle analogy the patterns would form in lines along the trestle points...

...and a phase period is how much time it takes for an up and down movement to complete...

...and phase shifting is when a phase period shifts from 1 timing to another.

In some instances the Chladni patterns themselves could be considered 3 d, in that more sand gathers in some areas than others causing raised contours and shading...

...Given the impossible scenario of gravity being an equal force on both sides of the plate, (ie: an equal measure of sand, as was placed on top of the plate, placed on the underneath of the plate, doesn't fall off the plate), the pattern would be the same on the bottom of the plate as it is on the top.

A simple form of a 3 d side view of this effect, in that a side view of a cube is a square, might be viewed as a horizontal and vertical cross section of equal lines, such as the lines that would divide a square into 4 squares, but with curves shaded into the innermost corners, making a 4 pointed star-ish sort of shape. Looking at the shape as its 3 d form, it would be a 6 pointed star with inverted curves between the points. 4 points on the horizontal and 2 on the vertical. We can fit 4 circles into the curves between the points on the horizontal plane, which is the plate, and now these points are joined by the arcs of these circles. We could fit 4 circles into the curves on the vertical plane on top of the plate, and 4 more on the vertical plane underneath the plate, and if we considered the 6 pointed curved star shape that we started with and were to call this original shape an inverse representation of something, by adding the circles to finish off the innermost curves, we would have constructed the something that that the six pointed star shape is the inverse to. Clearly adding the circles to the vertical plane in the case of the plate only has meaning in respect to air displacement, its the activities on the horizontal plane of the plate that are displacing metal, but you can get the idea.

Also in watching YouTube videos of close ups of sand moving on Chladni plates, a very similar scenario to Lissajous figures that appears to be moving in 3 d occurs, and can be physically observed of movement in the sand at certain frequencies.

But the maths for Chladni plate patterns and how the distribution of resonant vibrations change when alteration to the phase period of the input signal are conducted already exist...

The maths for Lissajous figures of of various input phase periods already exist...

And although Lissajous and Chladni patterns are not the same patterns, they are both created by altering phase periods and I'd like to know what the mathematical difference is between how these patterns are being created by these phase shifts.

Looking at how the light of the laser beam is bounced of a vibrating mirror at right angles onto another vibrating mirror that bounces the light off at a right angle onto a screen - when considering that a laser beam bounced off a vibrating mirror on the z axis, (which I'm understanding is the angle that is a straight on view) a Doppler shift will occur.

But the light being reflected at a right angle will not be Doppler shifted by the vibration. Something else is occurring... And given that the next mirror that reflects the light onto the screen at the opposite right angle is vibrating at the same frequency, the exact and opposite effect is added to the beam of light again before it hits the screen.

Isn't this basically adding a right hand slanted half portion Doppler shift to a left hand slanted half portion Doppler shift?