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There are lapping plates with a diagonal pattern, but squares seem prevalent.
Aside imparting a shearing action, or possibly supporting narrow items better, not convinced any advantage one over the other.
Smaller plate, narrower the grooves though, is logical.
re the harmonious blacksmith and .0001 achievement is correct. Development of metal planers kicked off in 1750's, concurrent among several inventors. Whitworth seems to have claim on the half-tenth with 3 plates, and applied to his planer, remains as first really accurate machine tool.
It's some kind of coincidence; a slotter, a planer, and a couple shapers appeared all the same day.
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Thanks N4NV! We've added your Lapping Plates to our Sharpening category,
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<a href="https://www.homemadetools.net/homemade-lapping-plates-4">Lapping Plates</a>
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