It's amusing when someone who can't read uses a condescending tone.
Next time try to comprehend what the discussion is about before chiming in. And a little tip - insulting people while trying to make a point just telegraphs your own insecurities, it doesn't advance the discussion at all. And keep in mind that when trying to show your superiority, it's not a good idea to assume nobody else has ever taken a degree. It's actually quite common. Just in my family alone I think there's 5 engineers. I'm not exactly sure who is impressed by that these days.
So lets summarize:
1) Friction in a dynamic system is complicated and there are many factors to consider. F=coefficient of friction * normal force is the best approximation we have for a static friction system and is purely a model with many assumptions.
2) Tires are deformable bodies and come in many flavors. To argue about contact patch you need to be using the same kind of tire. In that argument the same kind of tire will not have a larger contact patch with a wider sized tire with the same amount of air pressure in the tire (run flats obviously don't fit this assumption and work under different rules). The difference is the shape of the contact patch with wider tires.
3) Contact patch is merely one factor of many in tire design. A general equation we have been arguing over is P=F/A (pressure of tire = force of vehicle / area of contact patch). It is a good approximation but obviously is tire dependant as different tires will deform in different ways and as you near the extremes of tire pressure it will become more inaccurate.
Now back to the root of the argument.
Wider tire =
better able to compensate for track imperfections
More tire to dissipate thermal energy to (you need a little temp, not too much)
Will greater sidewall increase your contact patch area? Maybe. Depends on the tire and the tire wall stiffness. You change the wheel size and the mechanics of the tire change also, even with similar types of tires. But to say the gains you get are from contact patch area secondary to a taller sidewall .... I would disagree when you compare 17" to 18" rims (the bigger the difference of the rims, at a lower pressure, you will see a more significant change). The biggest factor in this 1" difference would be tire pressure. The gains from that increased sidewall are probably most significant because of weight savings. That being said, you probably have a very small increase in contact patch area with a bigger side wall and it does have a very minor role in increasing traction. However I would say that the saved rotational mass would be a more significant factor here.
And all that being said, the main point of everything is that tires are a balance of lots of different factors. The best way to measure all of these is to go out and test them in the environment. So with time, drag racers have used experience and indirectly physics to go faster. Low weight, big contact patch, thermodynamics/heat transfer issues means finding the right balance of sidewall height, tire width, stickiness of rubber compound / softness of the compound, and.... the driver to do it all
This is a great topic that lots of people have questions about and there is probably a thread created everyday about tires or something about traction. There are lots of rumors and pseudo science about this topic. I would love to keep talking about it and come to a good consensus about it. I love looking things up and I'd love to learn and teach things if I can.
Tires are deformable bodies ... different tires will deform in different ways ... Will greater sidewall increase your contact patch area? Maybe. Depends on the tire and the tire wall stiffness ... you probably have a very small increase in contact patch area with a bigger side wall and it does have a very minor role in increasing traction ... The best way to measure all of these is to go out and test them in the environment. So with time, drag racers have used experience and indirectly physics to go faster. Low weight, big contact patch, thermodynamics/heat transfer issues means finding the right balance of sidewall height, tire width, stickiness of rubber compound / softness of the compound, and.... the driver to do it all
Again, throwing the physics out the window and using only experience... drag racers use the tallest widest tires with the biggest sidewalls the rule makers will allow. And sports car roadracers opt for the smallest diameter and widest wheel that will clear their brakes. In both cases, their wheel & tire choice is not dictated by budget limitations, but rather the desire for maximum traction. Form follows function.
Incidentally, I love this picture. It's me exiting turn 17 at Sebring International Raceway a couple weeks ago. I'm sitting on 275/40-18 NT-05s at about 40psi hot at that point. I was surprised by how much the sidewalls deformed...
And check out these videos of sidewall deflection on autocross cars:
I'm not trying to make a point one way or the other, besides saying that tires almost act like mushy balloons when you pay attention to them. Treating them like rigid bodies misses A LOT of what tires actually do.