Push-up calculations: Feline facilitation, quantified

Not everyone thinks of cats as being big helpers, but mine seems to very much enjoy helping me with my workouts — with push-ups especially:

push-up partnerSome might say that she is actually hindering the doing of push-ups, which is technically true . . . but since the whole point of the exercise is, well, to exercise, the extra weight is ultimately helpful.

And it actually makes a difference! At least, it feels like it makes a difference. Despite her affinity for cookies (among other things), she only weighs about nine percent as much as I do; yet, push-ups feel more than nine percent harder when she’s up there. Is that just my perception? Or is there something about how the weight is distributed that gives her contribution a disproportional effect?

I decided to investigate the matter, breaking it up into three key questions:

1. How much force is actually required to do a push-up?

2. How does the addition of a medium-sized feline passenger increase the required force?

3. How much does additional force change the difficulty of the exercise — or, relatedly, the number of push-ups that one should expect to be able to do?

Continue reading

Cookie calculations

Every time I bake cookies, I wonder if I’m making the most of the space on the cookie sheet. I finally decided to stop wondering and figure it out.

Experimental methods

My investigation involved circular cookies of two different sizes:

cookie sizesTwo examples of large cookies and three examples of small cookies. There was some standard deviation in the size of each group, but it wasn’t too bad.

They were arranged on cookie sheets of two different sizes:

cookie sheet sizesThe sides are, in fact, straight. It’s just the camera that makes them look curved.

Now, the most common way to arrange cookies is in equally spaced rows and columns. This is called a square lattice; when you draw a line from the center of a cookie to each of its four “nearest neighbors,” you end up with a pattern of squares.

Another option is, instead of putting the cookies in row two directly under the cookies in row one, to arrange them in the “gaps” between the cookies on the first row. If you do this, you get a triangular lattice, also called a hexagonal lattice; when you draw a line from the center of a cookie to each of its six “nearest neighbors,” you get a pattern of triangles.

proto-cookies in square latticeproto-cookies in triangular latticeCookie dough arranged in a square lattice (OK, so this one is a slightly rectangular lattice, but you get the idea) versus a triangular lattice.

As you can see, the cookies are more densely packed when they’re in a triangular lattice, even though they’re the same distance away from the cookies closest to them. If you work out the geometry problem, it turns out that cookies arranged in a triangular lattice take up 86.6 percent as much space as cookies in a square lattice. In other words, you can fit 15 percent more cookies into the same amount of area.

Well, problem solved! Triangular lattice it is.

But wait! What if your cookie sheet isn’t wide enough for an offset second row? Is the triangular lattice still the way to go? Is it still worth it to get the rows closer together, when it means every other row is a cookie short?

To find the answer to these questions, I first tested the four different set-ups in my kitchen (small cookies on a small cookie sheet, small cookies on a large cookie sheet, large cookies on a small cookie sheet, and large cookies on a large cookie sheet). Then I used equations and graphs to generalize the answers.

Continue reading