Hello Dusty,
just for clarity:
Boyles law states that: the pressure of a given mass of an ideal gas is inversely proportional to its volume at a constant temperature
I cannot deny that temperature changes will occur in working tyres, but to try and keep the explanation as simple as possible concerning the effects of volume on pressure I deliberately chose to remove them from the discussion hence my comment in my posting "ignoring temperature changes" meaning to infer constant temperature. I considered it reasonable because they will be relatively small compared to the effect we are discussing.
Your reference to the footprint is very pertinent, and over or under inflation is bad news, but within the tyre manufactures remit must be a remit to ensure that for any given pressure the tyre will continue to perform satisfactorily with a range of applied loads. were not talking about F1 edge of working limits here. But I did use the concept of the "contact area" which is the same as the foot print.
As an area is defined as its width times its length, if one dimension is constant (i.e width) then the change in length is directly proportional to the change in area. Again, I deliberately chose not to vary the width of the tread because it would over complicate the point I was making. In practice the tread width does not change very much once the pressure has been set so it was I believe reasonable to consider it a constant in this principle.
I do concede I could have described the tyre slightly differently:
Looking at a side view of tyre on any vehicle it has two distinctive parts to its circumference. There is flat portion where it is in contact with the road, and the remaining part which is a segment of a circle. Bearing in mind my simplification that the width of the tyre does not change, the the area of the side of the tyre is proportional to its volume.
Considering the same tyre but with different applied loads, if you increase the load the contact area will increase which will be represnted by a lengthening of the contact area and to accommodate this the axle has to drop accordingly, thus the side wall area of the tyre has reduced, which in turn means the volume has decreased. Due to Boyles Law, that means the pressure will also have risen but not in proportion to the load but in proportion to the change in tyre volume.
Importantly if you double the load, you don't lose half the volume of the tyre, only a relatively small proportion, which is why the pressure changes are also relatively small. Boyles law applies.