Visualizing relationships, be they mathematical, cultural or physical is a powerful skill. Here is a great site about an interesting quest for visualizing prime numbers.
I am a big fan of Phi, the Golden Ratio. One of the best books on this is Mario Livio's The Golden Ratio. Livio is a serious mathematician, historian and story teller who brings alive the beauty of seeing how things relate. The golden ratio shows up in many optimal packing and density solutions involving complex systems, it abounds in nature from the circling of an attacking hawk to the positioning of leaves on stems.
My guess is we haven't scratched the surface of applied uses for constants like Phi or Feigenbaums constants. For those interested in how growth and form emerge from nature and complex systems I would suggest studying morphogenesis, starting with the classic On Growth and Form.