(New for school year 2015-2016)
This video shows a series of (logarithmic) spiral forms in nature as an opening to the study of all kinds of patterns the teacher could introduce. A good overview of the spiral and patterns in general can be found here: https://en.wikipedia.org/wiki/Logarithmic_spiral (note list of natural phenomena) and https://en.wikipedia.org/wiki/Patterns_in_nature
There of course are natural forms that replicate basic geometric forms like cones, lozenges, ovals, and spheres. But more complex ones are often built on hidden formulas, which are hinted at if you look for them. For instance, this leaf has a 3-pronged overall design, and the main division of the leaf in turn reflect the same pattern, called "iterations." http://i80.photobucket.com/albums/j177/lesmuskey/af530582-748d-4de4-8933-1f6f079598b9_zps9qa9pqgv.jpg
Now that computer imagery can express formulas many times into a fantasy-like geometric image, and then allow us to zoom in to ever more magnified views which no leaf could follow in practical life, such formats are called "fractals." Students can find many of these shared on the internet.
Once your class starts looking for them, they will find patterns everywhere and can speculate and discuss which ones may or may not be fractals. But at least they will appreciate the structures of living and inanimate things and enter the speculation of whether there are underlying resonances through the universe.
A note of caution: The popular "golden rectangle" and "Fibonacci" forms have generated almost a euphoria and finding them to possible excess, and this has its skeptics, which I cite in the links below. Also, you might get students to realize that while we can see some undeniable patterns, we can also imagine them when they are not based on mathematical formulas, but just chance. Examples would be the many sightings of "faces" in clouds, in tree barks, and even in such things as french toast. I saved this photo of the cracking pattern in aging asphalt. While parking lots produce many similar "nets," it is doubtful there is any operating cause here that would really make the cracks form a spiral, not even turning wheels, since it was in the center of a parking space. http://i80.photobucket.com/albums/j177/lesmuskey/asphaltspiral_zpsdv2pndqp.jpg
Further comment: I tried to not use slightly exotic examples such as agave, but also left out some items that you can easily find and bring into class (some having multiple spirals) such as acorn tops, fresh pineapple, pine cones, and even sliced red cabbage and opening rose blooms. Cactuses as a group have many fascinating samples also.
Following the fungi, begonia leaf and fern, the seed/plants sequence in the film shows: poppy ovary inside, medicago seed, prickly caterpillar pod, clematis fruit, unknown aster, Queen Anne's Lace, 2 wildflowers similar to phacelia or comfrey, a moonflower, a sunflower center, a hibiscus, a gardenia, and 2 opening stages of datura. I also used small animals, but some believe there are complex spirals on the human body, notably the outer ear, and certainly in the horns of large animals such as mountain sheep, spiral horned antelope, and water buffalo....and even how elephants curl their trunks..
Another fascinating thing I noticed, that I didn't see comments about in my readings, is that the origin point of the spirals in shells is the center, with outward expansion taking place by the addition of chambers as the animal grows within. On the other hand, the form of flowers such as blue phacelia curl inward as they grow, meaning that in a series of buds, the oldest ones open on the outer curls and the new unopened ones show to the center of the spiral. A photo is here: http://i80.photobucket.com/albums/j177/lesmuskey/bluephaceliaorig_zpsk2xcb8db.jpg
If you cover branching patterns, that will lead to rivers and body circulation in animals. If you cover spheres, have students notice the "wasted space" between them as they crowd is eliminated if they turn into efficient hexagons as in the wax combs of a beehive. The first image of an ammonite is actually of an extinct sea creature in fossilized form, which is a fascinating topic unto itself.
Finally, a list of websites for inspiration:
https://www.pinterest.com/explore/spirals/ (art and nature examples)
http://fract.al/background (very advanced)
http://skeptoid.com/episodes/4325 (good explanation of spiral growth helps eliminate leaves shading each other; more doubting of human body and architectural claims)
http://nautil.us/issue/0/the-story-of-nautilus/math-as-myth (for adults: universe is messier, but essay is sympathetic to the search for order)
See other films by DeepRiverVisions at https://www.teacherspayteachers.com/Store/Deep-River-Visions