In the classical system of the five orders of architecture, even the most complex or intricate elements can be specified by numerical patterns of modules and parts. Such patterns, known as canons of proportion, have come down to us in various architectural treatises which designers may use as references. This is a rational approach to achieving beauty, an Apollonian path, fully visible and based in text. But we might remember that Apollo also follows another path to beauty, as leader of the Muses and master of the lyre, an instrument he received from Hermes and gave to Orpheus. In this article we will explore this less traveled though no less worthy path, which is more often associated with Orpheus, Dionysus or Hermes than with Apollo. This path is not irrational but rather hyper-rational, one that includes the rational and more, what might be called mythopoeic or even ecstatic. Such a journey is justified not on romantic or hedonistic grounds but is necessary to explore inner, hard-to-see aspects of number.