"Mathematics is the alphabet with which God has written the universe." -- Galileo Galilei
Nature holds many evidences of God’s creation. One of the evidences of God’s creation inscribed in nature is mathematical principles. The well-known mathematical principles found in nature are golden ratio, golden angle, golden rectangle, Fibonacci sequence, logarithmic spiral, and fractal. Let’s first briefly describe what these mathematical principles are.
When the ratio of the larger length to the smaller one is equal to the ratio of the sum of the larger and smaller lengths to the larger one (a/b = (a + b)/a if expressed algebraically (see Fig. 23)), it is called golden ratio. The golden ratio is approximately 1.618 (1+sqrt(5))/2). The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio (Fig. 23). The approximate value of the golden angle is 137.51 degrees. The golden rectangle is one whose side lengths are in the golden ratio.
The Fibonacci sequence is the series of numbers where the next number is found by adding the preceding two numbers (ex: 1, 1, 2, 3, 5, 8, …). The logarithmic spiral is the one within which any line emanating from the origin cuts the curve under a constant angle (see Fig. 24). The fractal is an object or quantity that displays self-similarity where a fragmented geometric shape is a reduced-size copy of the whole. With these mathematical knowledge in hand, let’s start to explore where we can find these principles in nature.
Fig 23. Golden Ratio (left), Golden Angle (middle), Golden Rectangle (right)
Fig 24. Logarithmic Spiral (left) and Fractal (right)
Leaves, petals and other organs in plants are arranged orderly and follow some mathematical rules. This geometric arrangement of organs is the main topic in phyllotaxis study. In about 80% of plants, leaves are arranged along the stem tracing a helix. Suppose we fix our attention to some leaf on the bottom of a stem on which there is a single leaf at any one point. If we number that leaf "0" and count the leaves upward along the stem until we come to the one which is directly above the starting one, the number we counted becomes one of the Fibonacci sequence.
If we count how many revolutions we made when we count the number of leaves along the stem, the number of revolutions also becomes one of the Fibonacci sequence. The arrangement of leaves can be expressed as a ratio of number of leaves to number of revolutions. For example, if the number of leaves in our sample plant is "5" and the number of revolutions is "2", then our plant is said to have phyllotaxis 2/5 (see Fig. 25). Each plant can be characterized by its own phyllotaxis. Almost always the ratios encountered are ratios of consecutive or alternate terms of the Fibonacci sequence (for example, 1/2, 1/3, 2/5, 3/8, 5/13, etc.; notice here that numerators and denominators are also Fibonacci numbers). The angle formed by two adjacent leaves remains approximately constant and approaches to the golden angle (which is 137.5 deg) as the number of leaves increase. If they have golden angle, all of the leaves in the plant receive the same amount of sun light and can collect water effectively and direct it to the center of the plant to pass down to the root.
Fig. 25. 2/5 phyllotaxis (top) and 3/8phyllotaxis (bottom)
Not only for the leaves, but also for the buds, seed heads, and shoots follow mathematical rule. If seed head has golden angle, it can have the largest number of seeds in a given area (sunflower seed is a good example). If we look closely to these shoots or seed heads that have golden angle, we can find spiral patterns (logarithmic spiral) as shown in Fig. 27. Here, we can find interesting properties in these spirals. The number of spirals is two or three adjacent Fibonacci numbers. For example, the number of spirals is 21 in clockwise direction and 34 counterclockwise direction in daisy flower, 8 clockwise direction and 13 counterclockwise direction in pinecone, 5 counterclockwise direction, 8 clockwise direction, and 13 counterclockwise direction in pineapple, 5 clockwise direction and 8 counterclockwise direction in cauliflower, and 13 clockwise direction and 21 counterclockwise direction in romanesco broccoli. If the number of spirals form Fibonacci numbers, they preserve the same shape while they are growing. For example, the shape of the daisy flowers is circular when they are small and keeps the same circular shape while they are growing. The mathematical principle behind having invariant shape is that the spirals having Fibonacci numbers form logarithmic spiral and anything growing in logarithmic spiral pattern doesn’t change shape.
Fig 26. Growth pattern of the Norway spruce shoot.
Fig 27. Daisy flower, pine cone, pineapple, cauliflower, and romanesco broccoli. Numbers of spirals are two adjacent Fibonacci numbers.
The growth pattern following the logarithmic spiral can be found not only in plants but also in auricle, cochlea, fingers, seahorse’s tail, ram’s horn, and nautilus, etc. (Fig. 28). There is very important reason for these to grow in logarithmic spiral pattern. If they do, they can keep the same shape when they were young and after they were fully grown. The cochlea in your ear has nice geometric shape for best hearing. If it changing shape while you are growing-up, you will have hearing problem. Likewise, if bones in your fingers don’t grow in logarithmic spiral pattern, you will have hard time to grasp objects with your fingers when you become a grown-up.
Fig 28. Pinna (left), nautilus (middle), and finger bones (right) are grow in logarithmic pattern
Fibonacci numbers can also be found in sneezewort (Fig. 29 left). If we draw horizontal lines through axils of the sneezewort, we can see a growing pattern of the stem and leaves. The main stem produces branch shoots at the beginning of each stage. Branch shoots rest during their first two stages, and then produce new branch shoots at the beginning of each subsequent stage. The same law applies to all branches (see middle figure in Fig. 29). Now, if we count the number of branches in each section, the counted numbers are all Fibonacci numbers. Furthermore, if we count the number of leaves in each stage, they also form Fibonacci numbers (right figure in Fig. 29).
Fig 29. Sneezewort (left). The numbers of branches (middle) and leaves (right) in sneezewort form Fibonacci numbers.
Another mathematical principle found in nature is a fractal. The fractal structure can be found in fern, trees, cauliflower, etc.. If we look fern closely, we can find that a frond from a fern is a miniature replica of the whole. The circulatory, respiratory, and nervous systems in human body also have fractal structure. Human circulatory system needs to provide fresh oxygen and nutrients to every cell in the body. To archive such a task, the blood vessel system has to be designed in fractal structure. If it has a fractal structure, it can occupy minimum volume but has a maximum area to reach every cell in the body. It also reduces strong blood pressure by evenly distributing it to every end of the blood vessel.
Fig. 30. Fractal structure in fern (fern) and romanesco broccoli (right)
So far, we have seen many examples of mathematical principles in nature. How can that happen? Does the plant know high-level math and arrange their leaves in Fibonacci sequence? Does the sunflower know college level mathematics and produce its seeds in the pattern of golden angle? Does the fern know high-level mathematics and produce its leaves in the fractal form? Or all of these are designed by God just like instinct in animals?
If we found a stone age arrow in the field (Fig. 31), we know that it was made by stone age man long times ago since it's shape is different than other irregular-shaped stones and can't be made by natural process either water or wind erosion. Likewise, if humans and animals that are incomparably more complex than the stone age arrow are living on earth, then their existence themselves tell us that they were created and not evolved by natural process.
"For since the creation of the world God's invisible qualities--his eternal power and divine nature--have been clearly seen, being understood from what has been made, so that men are without excuse" (Rome 1:20).
God created all of these living creatures on earth!
Fig. 31. Stone age arrow. The arrow was created by stone age man long times ago.
"For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.” (John 3:16)