Tomatoes and spheres

Crossposted at Dailykos.com

For those of ye who do not know me well, gardening is sort of a release and a creative direction for me. For those of ye that do, the same. I am inspired by the by the bounty that my garden is now providing, and some of that bounty consists of fresh, ripe tomatoes, in all sizes. That, like everything does, got me thinking.

Store tomatoes look pretty, but are of little flavor and practically no “nose”. Farmer’s Market ones are much better, but often blemished. I will trade blemishes for taste and nose any day. Part of this has to do with commercial varieties versus small patch varieties. Commercial ones are hard, and mechanically picked underripe so they can stand transportation. Small patch ones are softer and often are left to ripen on the vine.

But the best are the ones that you pick yourself, daily, at the peak of ripeness, from either a garden (I am lucky to have enough space for one), or from a patio planter. Anyone that has full sun for at least half the day can grow them.

Tomatoes come in an extraordinary variation of varieties. There are red ones, pink ones, yellow ones, almost black ones, and of course, green ones of every variety. There are vine ones, cherry ones, big ones, middling ones, and small ones. Do the math, including the odd shaped ones, and you will quickly come up with a permutation in the hundreds of thousands, if not millions. Now they are breeding hard, marketable ones that are roughly cubic, for more efficient packing into shipping cartons. I kid you not, shape for shipping reasons is becoming increasingly important. Taste is less important, I suppose.

Tomatoes are uniquely American. For a long time, it was thought by the Europeans that only Native Americans could eat them and live, and they were planted purely as ornamentals in Europe for some time. Then someone decided to taste one.

That made the interest in tomatoes expand rapidly, and since they have been regarded as one the most succulent of all of the “vegetables”, although, technically, they are fruits. That does not matter, they are just good.

Now for the science part (everyone knew that it would come to this, I suspect). Tomatoes are mostly water, like most food, but have little starch. They do have a lot of sugar, and that factor should be recognized by folks suffering with diabetes. However, all modern diabetic diets recognize the importance of healthy fruits and vegetables, so this, while important, is not a problem.

The figures wander all over the place insofar as nutrients go, but generally, the redder the tomato, the more lycopene (an obscure but important nutrient) it has. All are full of ascorbic acid (Vitamin C), and in Vitamin A as well. I eat them, though, because I enjoy them. At my peak, and their peak, I can eat a lot of them.

They are also very rich in potassium, with little sodium. Potassium is an essential element for proper neural transmission, and many people on blood pressure medications (the diuretics) need supplemental potassium. I can think of no better way than from fresh, vine ripened, tomatoes, although I often add extra sodium in the form of salt.

Now for the next level of the science (actually, in this case, the math). My grandmum and mum always peeled tomatoes from the garden, because the peel is sort of hard to digest, and we, except for a single modern bathroom for all of us, had outhouses. The peel tends to “work” you, and for this reason it was often discarded. That is a pity, because lots of nutrients and dietary fiber reside there.

For this discussion, please allow us to consider a tomato as a perfect sphere. The volume for a sphere is V = 3/2*pi*r^3, where r is the radius. For surface area, the formula is A = 1/2*pi*r^2. I know that is is a bit difficult, but follow me here. When the volume is divided by the surface area, the constants 3/2 and 1/2 collapse to 3, and pi is eliminated. Thus, the relation between the volume of a sphere and the area of it becomes simply 3r, or three times the radius of the sphere. This hold ONLY in the case of the skin being infinitely thin as compared to the fruit volume. In other words, the volume of a sphere is 3r times the surface area. As r approaches zero, the tomato becomes all skin, and it approaches infinity, the tomato approaches all meat.So here is the problem: given two tomatoes, a little one and a big one, which has more “meat” as opposed to skin?

Let us assume two tomatoes, identical except for one is a cherry one, 2 cm (just under an inch) in diameter, and the other one just 20 cm (about 8 or 9 inches) in diameter, both with a 1 mm thick skin. If anyone can provide a sketch of these models, I would appreciate it. My freehand is not very good.

The relative volumes for each are governed by the exactly the same formula:

V (total) = 3/2*pi*(r)^3. Plugging in numbers, this becomes 3/2*pi*(1 cm)^3 so V (total) = 4.71 cm^3.

Likewise, V (meat) is 3/2*pi*(0.9 cm)^3. This gives V (meat) = 3.44 cm^3.

V (skin) is just the difference between the total volume and the meat volume. V (skin) = V (total) – V (meat), so, since we have already worked out those, V (skin) = 4.71 cm^3 – 3.44 cm^3, to V (skin) = 1.27 cm^3.

Thus, the per centage of skin to meat in a cherry (2 cm) tomato is given as % difference = ((V (meat) – V (skin)) / V (meat)) * 100, or, to use the numbers, ((3.44 cm^3) – 2.7 cm^3) / 3.44 cm^3) * 100, or otherwise to say that 21.5% of a cherry tomato is skin.

For the large one, the formulae are exactly the same, just the numbers change. Plugging those numbers in for a 20 cm tomato gives us the following:

V (total) = 3/2*pi*(10 cm)^3 so V (total) = 4712 cm^3.

V (meat) is 3/2*pi*(9.9 cm)^3. This gives V (meat) = 4572 cm^3.

V (skin) is 4712 cm^3 – 4572 cm^3, which gives V (skin) = 140 cm^3.

The per centage of skin to meat in a large (20 cm) tomato is then ((4572 cm ^3 – 140 cm^3) / 140 cm^3) *100, so in a large (20 cm) tomato, 3.1 % of of a large tomato is skin.

What does this really mean? Not a lot, I guess, but I find it interesting, both from the mathematical standpoint, but from the canning standpoint. When you can tomatoes, protocol requires that the skins be removed. My grandmum always complained about the “little, knotty” ones taking too much time to skin and get ready to process. Now we know why.

In summary, a cherry tomato is 21.5% skin, and a big one is 3.1% skin. No wonder the bigger they are, the faster they are to process, since each skin, at least at home, has to be removed by hand. In addition, the waste is much less for the larger ones, but I think that she was more concerned with the labor.

Finally, how do our cube-shaped ones act? For comparison purposes, let us assume a big one (they are not) with a 1 mm skin (generally thicker, so more waste, but we will only look at the geometry. Our tomato is a 10 cm cube, with a 1 mm skin.

The volume of a cube is given by V = s^3, where the s is the length of a side. The area is given by 6*s^2, since there are six sides to a cube.

So, the total volume is V (total) = (10 cm)^3, or 1000 cm^3.

The volume of the meat is V (meat) = (9.9 cm)^3, or 970 cm^3.

Then the volume of the skin is V (skin) = 1000 cm^3 – 970 cm^3, or 30 cm^3.

So for a 10 cm sided tomato, the skin is given by %difference = ((970 cm^3 – 30 cm^3) / 970 cm^3) *100, or 9.7 per cent skin.

Thus, “square” tomatoes always have more skin to meat than “round” ones.

I know that this is a little off topic, and perhaps even esoteric. I will hang on for a while for questions, comment, and such. Warmest regards, Doc.

Tags: Gardening, Geometry, Learning, Math, Skin, Teaching, Tomatoes