Way back on page one Mills75 mentioned the importance of symmetry to beauty, at least in regards to mate selection. It's been discovered that symmetry is a very important factor in the selection of a potential mate, and not just in humans, apparantly in many, if not all, species of animals. The factor of symmetry in mate selection was first studied in scorpionflies.

It was found that female scorpionflies preferred to mate with males with the most symmetrical wing lengths—right and left.
Female birds select the males by their beauty of their plumage and song. We now understand that the condition of the plumage reflects the genes, that is, the ability of the male bird to get food, resist parasites and disease, and, in general fend for himself, all desirable qualities to pass on to one's offspring. Now, the female bird—and the female scorpionfly too, I suspect—doesn't consider the male's plumage and think about his genes; she just thinks, "wowsa! I want him." To her he's beautiful. Essentially, I believe it's the same for all animals in regards to mate selection.
Humans have the ability to perceive beauty outside of mate selection, and we may be unique in this. As others have suggested, the mystery of beauty will always remain so, although it probably can be killed with over-intellectualization, just as the mystery of poetry and music can be killed by over-analyzing them.
With that in mind, and since the thread began with a photo of blossoms, I'll give a short lesson on the flower's reason for being. We must remember that showy flowers began, that is, came in existence, to help potential pollinating insects find them. The shape, color, and odor of flowers—the same factors that attract us—exist to attract the insect and act as signposts in contrast to the bland and mixed-up background of leaves and such. The symmetry and color of the petals allow the insect to see the flowers in a sea of leaves. Insect sight is something like our sight when standing too close to a mosaic on a wall; not that good and bold symmetry and color make it easier to pick out patterns. The insect, rewarded with a meal of nectar and pollen remembers that particular flower species and is wont to return to other members of that species, and, in the process, transfers the pollen.
The insect is not interested in the beauty of the flower, but in the contrast with the background, perhaps not unlike our own response when walking through a prairie of monotonous grasses and coming upon a showy flower. We appreciate the beauty of the single flower in contrast to the thousands of identical grass plants. But should that same species of flower pop up in our backyard bed of tulips, we may pull it up as a weed.