What is synthetic geometry? Could you provide a short (i.e. a paragraph or two, not much longer) explanation in general elementary terms? In particular, I hope to be able to understand the contrast between synthetic and analytical (invariant-theoretic?) approaches in the end.
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$\begingroup$ Have you tried Wikipedia? $\endgroup$– Ian ColeyCommented Feb 9, 2014 at 1:00
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$\begingroup$ I assume you want an explanation that differs from wikipedia. Or perhaps wikipedia is too advanced? Please be more specific. $\endgroup$– RghtHndSdCommented Feb 9, 2014 at 1:00
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1$\begingroup$ @rghthndsd Wikipedia's definition is too advanced. I don't really understand the difference between analytical and synthetic as it is discussed in that article. $\endgroup$– bzm3rCommented Feb 9, 2014 at 1:01
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$\begingroup$ do you know how to do geometry with coordinates? $\endgroup$– janmarqzCommented Feb 9, 2014 at 1:08
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$\begingroup$ "Synthetic" just means axiomatic in this context. $\endgroup$– goblin GONECommented Feb 9, 2014 at 1:49
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1 Answer
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Depending on the education you've had, the difference is likely basically that between your school algebra and geometry courses. In a first geometry course you usually get a list of axioms for points and lines (a straight line segment can be drawn between any two points ,etc.) Then you create proofs of properties of figures in the plane using these axioms, without ever writing down any algebraic equations. Specifically, you never use coordinates on the plane in doing this sort of geometry, which is what is called synthetic.
The ancient Greeks took synthetic geometry in the plane and 3-dimensional space to an amazing level, much further than almost anyone learns today. For example, they understood the "conic sections," that is, parabolas, ellipses, (circles,) and hyperbolas, almost completely. But today you usually think of a parabola, for instance, as being the graph of an equation $y=ax^2+bx+c$. It's this equation, which requires choosing a pair of coordinate axes on the plane and a unit length on each axis, that's what analytic geometry uses that synthetic geometry avoids. The synthetic-analytic distinction is still very visible in modern geometric research, even though the particular questions of interest have changed drastically.
answered Feb 9, 2014 at 1:08