This time of year finds me at my cabin in the Upper Peninsula surrounded by nature. And appreciating nature is enhanced by understanding the underlying mathematics and physics.

For example, Einstein was once walking on a beach with a friend and asked him, “Why do our feet sink into the dry sand, and also into the sand under the water, but the moist sand at the water’s edge holds firm?” Einstein then explained that the water molecules in the moist sand form a molecular table-top of sorts giving rise to surface tension. Dry sand has no such water, and sand under the water has water above it as well, so there is no “table-top.”

This is also why I see lots of spiders and other bugs skimming along the surface of the water when I kayak.

Here’s another one: Why do the waves easily move sand about, but larger stones and boulders remain still? “They are heavier” is a ready reply, but whether the rock moves depends on the rock’s weight compared to the force of the moving water on its surface. As the rock grows, both its weight and its surface area increase, but (here’s the key!) the weight grows more than the surface area.

This is apparent for anyone (such as I) who built things with square blocks as a child. Suppose you have a bunch of one-inch cubic blocks. One block by itself has a volume of 1 cubic inch, and a total surface area of 6 square inches. Agreed? So the ratio of volume to surface area is 1:6.

Now let’s make the cube bigger: 2 by 2 by 2. The volume is now 8 cubic inches, and the total surface area is 6 x 4 = 24 square inches. So the ratio of volume to surface area is 8:24 = 1:3.

What happens when it grows to 6 by 6 by 6? Now (calculate), the volume is 6 x 6 x 6 AND the surface area is 6 x 6 x 6, so the ratio of volume to surface area is 1:1.

One more: A 10 by 10 by 10 cube has a volume of 1000, and surface area of 600, so now the ratio of volume to surface area is 1000:600 = 5:3.

See what is happening? As the object grows (keeping the same shape) the ratio of its volume to its surface area increases. That is, although they both grow, the volume grows more than the surface area.

So, as a rock increases in size keeping the same shape, its volume (hence its weight) grows faster than its surface area — on which the water pushes. Eventually, the weight-to-push ratio becomes so large that the boulder doesn’t budge.

Another example: Why do elephants have big ears? Every cell in a mammal’s body is a little furnace, giving off heat. So the amount of heat produced by a mammal is proportional to its overall volume.

Mammals release heat through their skin that covers their body. So the amount of heat released is proportional to the animal’s surface area. For mammals of human size, the ratio of generated heat to released heat is about equal. But for large elephants, the generated heat due to their volume has overgrown their ability to shed their heat through the surface area. Thus, they need some extra surface area (without much volume). Hence the large floppy ears.

Understand? Here’s a nature question for you to answer using the same principle: Why do our cells divide as they grow? Why do they stay so small?

Let me get you started. Our cells are living things that need food and oxygen to live. These are gained passively by letting them soak through the outer cell wall. There is no active pump, they just seep through. Thus the amount of nutrients and oxygen that a cell gets depends directly on its surface area.

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On the other hand, since everything inside the cell needs food and oxygen to live, the amount of food and oxygen that a cell NEEDS depends directly on the volume of the cell.

Can you take it from here? A small cell can get its food and oxygen through its cell wall at a sufficient rate to feed all the living material inside. But as it grows, the volume (hence the need) grows faster than the surface area (supply), so the cell divides to reestablish the correct ratio.

Neat! Go forth in wonder!

— Community Columnist Tim Pennings is a resident of Holland and can be contacted at timothy.pennings@gmail.com. Previous columns can be found at timothypennings.blogspot.com.