How Small is the Point of Contact Between a Perfect Sphere and a Perfectly Level Surface?
Euclidean geometry suggests that a point is the smallest measurable object, with a size of zero. Any non-zero size would imply the existence of an area, which is unacceptable in this context. In reality, the point of contact between a sphere and a perfectly level surface can be considered theoretically as a single point, but the physical reality of contact is more nuanced.
Physical Reality of Contact
The relative hardness of the mating surfaces plays a critical role in the contact area. For instance, in hardness testing machines such as the Brinell hardness tester, a hardened steel ball of 10 mm diameter is pressed into the material to be tested, and the depth of indentation is measured to determine the hardness. In the Rockwell hardness tester, a diamond tip of known radius is used.
Even when considering only the self-weight of the sphere, the contact area can be determined using the modulus of elasticity of the materials involved. For a rubber ball on a cement surface, the ball will flatten depending on its weight, rather than maintaining a fixed diameter.
Philosophical and Theoretical Considerations
The concept of contact is somewhat philosophical in nature. In real life, many physicists might argue that nothing truly "touches" or "contacts" anything else. In an idealized 'geometric world,' the issue of contact is not straightforward. If two non-parallel lines cross, there is a single point of contact. However, in a three-dimensional space, the situation becomes more complex. A sphere resting on a flat surface might share a single point of contact, or it might not, depending on one's interpretation.
If they share a single point, it's akin to the metaphor of conjoined twins; lifting the sphere would require ripping a point away from the surface, or the sphere itself. If they are not sharing a point, then the concept of "contact" becomes ambiguous; it could mean that the sphere is hovering above the surface, leading to a non-zero distance between them.
Imagine a scenario where the point of the sphere closest to the surface is 1 mm above it. This point can be moved closer, but can it ever reach zero distance without sharing a single point? The answer is no, as a point still has zero size and share no space, meaning any contact point is, by definition, infinitesimally small.
Conclusion
The point of contact between a sphere and a flat surface can be theoretically considered a single point, with no size or zero size, aligning with Euclidean geometry. Understanding the physical and theoretical aspects of contact is crucial for various applications, from engineering to theoretical physics.