Surprise, outrageousness. An immediate confrontational tension is created. The response to a paradoxical move might be 'How can this be possible?' or 'That simply cannot work!'.
An example would be the move 2 Qf2!! in the solution to the Gurvich study (see the first example in the book) or the underpromotion 6 c8R!! in the Saavedra position.
To win by such means is a heroic form of achievement, and, other things being equal, the more paradox in the play, the better.
Subtlety, complexity. A deep move is one which is not obvious (though not necessarily paradoxical) and for which the point is well hidden. Initially one does not understand it, and later the response is 'Ah, so that was the point!'.
In the study by Ratner given earlier, the move 1 Be2! would qualify as being deep. In the Richter position 1 Kb7!! is both paradoxical (moving away from the action) and deep. Depth relates to the complexity of what is being achieved. Again, other things being equal, the deeper the moves the stronger the aesthetic effect. A game with no deep moves at all might be enjoyable the first time you see it, and maybe even a second time if it is good for other reasons; but some degree of depth is required to generate sufficient tension to count as a true masterpiece that can be played over many times with pleasure.
In our extended meaning of 'geometry', any striking pattern or special feature could be included. For example, if during the solution to a problem one side promotes pawns to each of the four possible pieces (Q, N, R, B known as 'Allumwandlung') this would be a 'pattern' or feature that the brain might easily recognize. Many tasks and special effects achieved by composers fall into this broader 'geometry' which is not restricted entirely to its spatial sense.
Patterns , repetitions, echoes, mutual interferences between a rook and a bishop... The response might be 'Oh, what a pretty pattern!'.
An example of mainly geometrical play would be the Nissl position (see the end of the brief history section). Here the bishop jockeys with the rook (in what could be described as a geometrical duel) before completing a diamond- shaped 'rundlauf' (Bg5-h4-g3-f4-g5).
In the Sam Loyd Organ Pipes mate in two, the mutual interference on lines and diagonals would also be called geometrical. As explained earlier, the brain is good at spotting such patterns and the prettier the pattern that is involved in achieving something, the better (the tension is resolved in a pleasing, aesthetic fashion).