# "The only plane that is common between three or more surfaces HAS to be flat." <-- How?

I was watching this video which made the claim in the subject line above. Unfortunately they do not explain it (there is a brief graphic but it made no sense to me).

The video is about extreme precision in tools made in the 19th century. Remarkable precision for the day using mechanical means. One of the things they managed were very, very flat surfaces which were achieved (it seems) via the method in the subject line.

But unless I am missing something obvious I do not understand what they were doing to achieve these really, really flat surfaces.

Can anyone shed some light on this?

Did you read the wikipedia article(s) linked in the description for that video?

If you scrape plate B to fit snugly with plate A, you may make B as flat as A. But if A is concave, B will be convex and vice versa.

If you scrape plate C to fit snugly with both A and B you will remove both the convexities aligned with A’s convexities and those aligned with B’s convexities. C will be smoother than either A or B. For best results, repeat this iteratively, using C to improve A, B; then use A and B to scrape C again.

Three plates, A, B, C.
Grind A against B, B against C, C against A, round and around until you reach a point where A is snug against B, B is snug against C, and C is snug against A.

Since we must invert one slab we know that a coordinate on on slab is ground against a different coordinate on the other slab, if we take a coordinate pair (x,y) on the bottom slab and flip the other slab in the y direction, the touching coordinate on the top slab is (x, -y). Thus we know, once the slabs are ground down that:

A[sub]x,y[/sub] - B[sub]x,-y[/sub] = 0
B[sub]x,y[/sub] - C[sub]x,-y[/sub] = 0
C[sub]x,y[/sub] - A[sub]x,-y[/sub] = 0

For any point (x,y) and its pair (x, -y), call the slope of the line between the points s.
On A the slope is s. On B the slope must be -s.
if the slope on B is -s, it must be s on C.
if the slope on C is s, it must be -s on A.
Thus s = -s, and s = 0 for all points p.
QED.

You will note that this works because we have an odd number of plates.

This is the standard procedure for scraping plates starting from scratch. Scraping is the process by which the flattest machine surfaces are produced. I have learned it a bit. Today the ready availability of high precision yet inexpensive granite surface plates makes it easier.

You ink the surface plate then lay your work piece on top and slide it a bit. This marks the high spots. Remove them. Repeat. You eventually end up with a work piece with hundreds of spots per square inch that are co-planar.

Obviously there is a lot more to it. The thickness of the marking ink, the direction and overlapping of scraped spots, etc. Classes to learn scraping are expensive running nearly \$1000 for a three day course. You end up with a finished straightedge of your own manufacture.

Dennis