Final Project, Math 105
Arielle Derby, Molly Curren, Maya Patnaik
4/26/1999
Guide to Flower Photos
For our project, we took digital photographs of various flowers and plants around campus, then found their different spiral patterns. We used Photoshop to draw lines to show their spirals. Ideally, we had hoped to find the parastechies of each plant and flower, (and also to come up with perfect Fibonacci numbers) but sometimes we found that the plants had irregularities. This could be due to increased periods of growth, or generative mutations.
In the course of working on this project, the parastechies became easier for us to detect, and we became able to predict certain patterns from flower to flower. In addition, we became more comfortable with using a digital camera and the Photoshop program. Here is a description of what we did with each plant – the process we went through – and you can see the end results in our photos.
Pic10zoom (Green plant): This plant has spirals going in all different directions. We know that it has a generative spiral. It wasn’t possible for us to draw the generative spiral, however, because the plant isn’t flat like the photo. Its spiral is only apparent in 3-D.
Echinocactus Grusonii: In this cactus we counted parastechies that went both clockwise and counterclockwise. White parastechies—14, Pink parastechies—13. Obviously, there is something strange with this one – either our method of finding the spirals was faulty, or the plant has its own numerical pattern. Clearly it does not fit in at all with the Fibonnacci sequence.
Gymnocalycium Izozoysii: Here we counted parastecies that went both clockwise and counterclockwise as well. When we divided the number of parastechies we counted in half, we saw that they were Fibonacci numbers. Red parastechies—16 (8), Black parastechies—10 (5)
Aeonium Arboreum: We had a hard time trying to count the parastecies because of the shape of the leaves. We tried counting the parastecies on several occasions and we came up with different numbers each time. Therefore, we decided only to look at the generative spiral.
Encephalartos Altensteini Zamiaceae: When we were looking at the cone we noticed that there were definite parastechy spirals but after a while we noticed that the spirals changed their path and were all haphazard. Therefore we concluded that this cone does have parastechy spirals but only till a certain point.
Palm Tree: Initially we took this picture because we thought that it looked pretty but when we looked carefully we saw that there was a very distinct generative spiral; due to the angle of the photograph, however, it was difficult to draw.
Mammillaria Alvilanata: The parastechies in this white cactus were easy to see and easy to draw. It is one of the few plants where we got perfect Fibonacci numbers: Black Parastechies—13, Orange Parastechies--21
Yellow Flowers: The yellow flowers were problematic because they were all irregular. Although we could clearly see where the parastechies were in all three yellow flowers, we could not get correct Fibonacci numbers despite several re-draws. If you look carefully, in several places you can see the point where the spirals become irregular.
Yellow #5: Black Parastechies—21, Yellow Parastechies--35
Yellow #6: Black Parastechies—21, White Parastechies--29
Yellow #7: White Parastechies—21, Black Parastechies--27
