In 'A course in Phonetics' P. Ladefoged writes:
If we consider vowels to be specifiable in terms of three dimensions, this implies that the cardinal vowels fall on a plane in this three-dimensional space, as shown in Figure 9.4.
But this vowels are not fall on a single common plane! If this plane existed, it would contain at least a, ɑ and ɔ by definition and would be something like this (we can construct a plane by three points):
(Sorry for inaccurate picture, but the main idea is shown)
As you can see, there is no way to place all of this vowels on the same plane. (I can assign numeric coordinates to vowels and prove it in a more accurate way by calculations if it's needed).
Am I wrong? Is P. Ladefoged wrong?
UPDATE: Mathematical proof for @GregLee
Statement: There is no plane that contains all of this eight vowels at the same time.
Proof: This plane, if it existed, would contain lines a-ɛ and ɑ-ɔ, but this lines are skew, so there is no such plane. Why are they skew? Because
- They do not cross each other, because
1.1. If they crossed, it would be either common point of line segments [a-ɛ] and [ɑ-ɔ], either of rays [ɛ, upwards) (part of line a-ɛ) and [ɔ, upwards) (part of line ɑ-ɔ), either of rays [a, downwards) (part of line a-ɛ) and [ɑ, downwards) (part of line ɑ-ɔ) because of same height intervals of elements of this pairs.
1.2. Line segments [a-ɛ] and [ɑ-ɔ] do not cross
1.3. Ray [ɛ, upwards) (part of line a-ɛ) does not cross with ray [ɔ, upwards) (part of line ɑ-ɔ), because
1.3.1. Roundness of ɛ is less than roundness of ɔ
1.3.2. Roundness decreases upwards in ray [ɛ, upwards) (part of line a-ɛ) and increases upwards in ray [ɔ, upwards) (part of line ɑ-ɔ)
1.4. Ray [a, downwards) (part of line a-ɛ) does not cross with ray [ɑ, downwards) (part of line ɑ-ɔ), because
1.4.1. Roundness of a is less or equals in comparison with roundness of ɑ
1.4.2. Roundness increases downwards in ray [a, downwards) (part of line a-ɛ) and decreases downwards in ray [ɑ, upwards) (part of line ɑ-ɔ)
- They do not lay in the same plane, because lines a-ɛ and ɑ-ɔ have derivatives of roundness with respect of height with different signs.