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Plainly, they knew what a circle and sphere were and also a square and cube; but did they discuss the idea that a sphere was analogous to a circle but in 3 dimensions or similarly the analogy between a square and a cube, again, that cube was a sort of 3 dimensional square?

My guess is that while they understood plane vs solid, they did not talk about 2 vs 3 dimensions. I am always surprised what arose 2000 or more years ago, if not in Greece then in China or India or later among the Arabs -- in just my lifetime, I believe we have majorly revised our understanding of when certain concepts were discovered, sometimes pushing back the date hundreds of years as I guess literature from non-Western countries is investigated.

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    $\begingroup$ From Plato's cave to conic sections, they had pretty sophisticated ideas about dimensional reduction, extension, and projection. The condescending framing of "how much did they understand" is begging lots of "but, then, why weren't we taught this?"s. $\endgroup$ Jan 11 at 21:48
  • $\begingroup$ @CosmasZachos: I am looking for the concept specifically of "dimension." I am the last person to condescend to the likes of Plato and Archimedes and Indians/Chinese/Arabs with whose names I am unfortunately not familiar. $\endgroup$
    – releseabe
    Jan 11 at 21:51
  • $\begingroup$ They certainly studied sections of solids by planes and contrasted surfaces to volumes of solids, as well as lines partitioning areas; how could they stay away from intuitive notions of dimension? $\endgroup$ Jan 11 at 21:54
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    $\begingroup$ Ancient Greeks so much had it that they assigned different kinds of "magnitudes" to lines, surfaces and solids, and Euclid never even multiplies magnitudes of different kinds, see Grattan-Guiness, Numbers, Magnitudes, Ratios, and Proportions in Euclid’s Elements. $\endgroup$
    – Conifold
    Jan 12 at 1:02
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The ancients did not have a general "concept of dimension", whatever it can mean. But they had good intuitive understanding, which is demonstrated by Euclid's "Definitions":

The extremities of a line are points

The extremities of a surface are lines,

etc. He did not add that "extremities of a solid are surfaces", but probably he understood this too.

In modern mathematics, we have several (not equivalent) concepts of dimension, and one of them (inductive definition of dimension in topology) actually gives a rigorous basis for these "definitions" of Euclid.

And certainly the ancients could only think of dimension taking only values 1,2,3.

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  • $\begingroup$ If Archimedes had assigned an integer to the difference between a square and a cube, you can bet he would have thought, What would a 4-D square look like. That is my bet anyway. But he did not think about numbers like we do. $\endgroup$
    – releseabe
    Jan 11 at 22:08
  • $\begingroup$ Where (in which book) did Archimedes assign an integer to the difference between a square and a cube? $\endgroup$ Jan 12 at 3:58
  • $\begingroup$ i did not say he had. $\endgroup$
    – releseabe
    Jan 12 at 7:56
  • $\begingroup$ I am trying to think how amazing someone of Archimedes' time calling a cube a 3D square would have been or to simply describe our space as being 3D. In some ways, it does not seem that incredible, actually, but I think it requires not so much an understanding of geometry but an understanding of integers that they did not possess. They used integers to count visible things, not attributes of space. But it also seems like they could not have been far from realizing that a plane figure has only two of the things a solid possesses three of. $\endgroup$
    – releseabe
    Jan 14 at 10:16

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