# Do cardinal vowels form a plane in 3D-space?

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.

The picture:

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

1. 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)

1. 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.
• It's an abstract plane, on the centroid of the saggital section. But in fact such diagrams are only intended to be two-dimensional; details like lip rounding, nasalization, pharyngealization, etc. use independent articulators, and therefore don't really appear on that trapezoidal tongue placement diagram independently. Apr 7, 2015 at 20:16
• @johnlawlerinexile, there is no plane in geometrical sense of the word that contains all of this vowels at the same time, isn't it? Apr 7, 2015 at 20:21
• Your figure does not seem to be accurately drawn. The line connecting ɔ and ɑ should slant toward the back of the trapezoidal solid, but you've drawn it as though it is parallel with the faces of the solid. You can't prove anything by drawing a sloppy figure. Apr 8, 2015 at 4:05
• @GregLee, I updated post with some nongraphical math Apr 8, 2015 at 7:25
• A plane can contain non-parallel lines, so from the fact that lines are not parallel, it does not follow that they are not co-planar. Apr 8, 2015 at 14:51