The King of Infinite Space: Euclid and His Elements by David Berlinski is an excellent work which discusses Euclidian assumptions, proofs, and key theorems. In his time, Euclid formulated some 400 theorems in deploying elemental logic.
Fundamental definitions include the idea that a line has length but limited breadth. Another important property of a line is that the extremities of lines are points themselves. Finally, the extremity of a surface is a line.
Berlinski discusses unique geometric forms like the cardioid. A penciled cardioid appears to be an imperfect circle. Practical examples of the cardioid include the
cardioid microphone, caustic surfaces in jars bounded by liquid surfaces and loop antennas.
Scientists and experts in mathematics from Newton to Einstein have used Euclid's axiomatic system and methods of proof to expand upon the existing body of human knowledge. In fact, many of the intricate geometric forms have been applied in modern engineering and geodome surfaces. For instance, the engineering designs of master practitioners like R. Buckminster Fuller incorporate many of the advanced Euclidian principles described by Berlinski.
The King of Infinite Space is an engaging read for people in formal academe, as well as the general public. A strength of the presentation is that the author has descriptive prose to satisfy the general readership and the formal mathematics to attract scientists and professional engineers. The presentation is easy to read and the exhibits have varying degrees of sophistication from the simple to highly complex and multi-dimensional.