Visual research problems in geometry

Solution 1:

Tristan Needham, Visual Complex Analysis, Oxford Univ. Press.
Needham cover

"One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. ... to replace our rich visual intuition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices—but his are interesting." —Ian Stewart

Here's one figure from the book, p.135:

You can almost guess the theorem from the figure: The two spheres $S_1$ and $S_2$ are orthogonal iff the two circles $C_1$ and $C_2$ are orthogonal.

Solution 2:

Discrete geometry (tilings, tessellations, packings, etc.) seems to be for you. If you like looking at nice pictures, see, for example, some of the questions of Joseph O'Rourke (at MathOverflow and at Mathematics StackExchange). (I first learned about him from his book Art Gallery Theorems and Algorithms.)

Is the matrix $A$ diagonalizable if $A^2=I$

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