The most important difference between plane and solid Euclidean geometry is that human beings can look at the plane “from above,” whereas three-dimensional space cannot be looked at “from outside.” Consequently, intuitive insights are more difficult to obtain for solid geometry than for plane geometry.
Some concepts, such as proportions and angles, remain unchanged from plane to solid geometry. For other familiar concepts, there exist analogies—most noticeably, volume for area and three-dimensional shapes for two-dimensional shapes (sphere for circle, tetrahedron for triangle, box for rectangle). However, the theory of tetrahedra is not nearly as rich as it is for triangles. Active research in higher-dimensional Euclidean geometry includes convexity and sphere packings and their applications in cryptology and crystallography (see crystal: Structure).
The-figure-illustrates-the-three-basic-theorems-that-triangles-areThe figure illustrates the three basic theorems that triangles are congruent (of equal shape and …[Credits : Encyclopædia Britannica, Inc.]
Proof-that-the-sum-of-the-angles-in-a-triangleProof that the sum of the angles in a triangle is 180 degrees.[Credits : Encyclopædia Britannica, Inc.]
The-formula-in-the-figure-reads-k-is-to-lThe formula in the figure reads k is to l as m is to n if and only if …[Credits : Encyclopædia Britannica, Inc.]
[Credits : Encyclopædia Britannica, Inc.]
Thales-of-Miletus-is-generally-credited-with-giving-the-firstThales of Miletus (fl. c. 600 bc) is generally credited with giving the first proof that for any …[Credits : Encyclopædia Britannica, Inc.]
[Credits : Encyclopædia Britannica, Inc.]
Bridge-of-AssesBridge of Asses.[Credits : Encyclopædia Britannica, Inc.]
Quadrilateral-of-Omar-Khayyam-Omar-Khayyam-constructed-the-quadrilateral-shownQuadrilateral of Omar Khayyam[Credits : Encyclopædia Britannica, Inc.]
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