RP: Cat in the Hat comes back and gets small

Note: The following is a repost from some time ago. Today is the officially day to celebrate Dr. Seuss, so here is my Seuss-related post. Enjoy.

In the second Cat in the Hat book (I think it is the second one), the Cat reveals that he has more smaller cats under his hat. They are labeled A - Z with Z being so small you can't even see. Question: What is the sequence of sizes for successive cats? How big would Cat Z be?

Here is the first picture that Cat reveals Cat A. It is not trivial to measure their relative sizes because they are in different positions. I drew two circles, one around each head and looked at the circle sizes.

So, Cat has a head that is 165 px tall and Cat A has a head 61 px tall. Let me call the total length of Cat = 1 c, then Cat A would be: (assuming the two cats have the same proportions):

If Cat B were the same proportion of Cat A that Cat A is of Cat B, then Cat B would be:

If all the rest of the Cats are of the same proportion, then I can calculate the height of Cat Z.

So, could you see Cat Z? If Cat in the Hat is 1.5 meters tall, then Cat Z would be 8.9x10^-12 meters. Could you see this? The most common thing to compare small visible things is hair. Hair can be around 50x10^-6 meters. So, Cat Z is WAY smaller than a hair. In fact, you could fit 5 million Cat Zs across one small hair.

But wait. There is more

There are other pictures of the Cats in Cat in a Hat's hat. Here is the next one. It shows Cat, Cat A, and Cat B.

Using the same technique as before, this gives Cat = 136 pixels, Cat A = 63 pixels, Cat B = 47 pixels. So:

Ok, so Cat A is a little different (I will assume that is close before - or close enough). Cat B, however, doesn't fit the pattern I used before. So maybe each successive Cat is not just 0.37 times smaller than the previous. I could explore this further if only I had more data. I do! Here is the next picture from the book.

This gives:

Ok. This is a little odd. It doesn't really disagree with previous pictures, but this seems to indicate that Cat A is not related to Cat in the Hat like B and C are related to A. B and C seem to indicate that each successive cat is around 0.72 times smaller. I wish I had even more data. POOF! I do. There is another picture.

So, I am going to say (except for Cat A), each successive cat is about 0.8 times smaller. If I still use Cat A as 0.37 times smaller than the Cat in the Hat (which is 1.5 meters tall), then Cat Z would be:

This is small, but clearly not too small to see. What if I go with a lower limit of 0.7? In that case:

Indeed smaller - around hair sized. Too small to see? I don't think so. I guess Cat in the Hat lied to the kids. He is such a liar. Maybe there really wasn't a Cat Z. Maybe Cat Z went to the New Zoo by Gerald McGrew, who knows. "hey look kids, here is Cat Z. I know you can't see him, he is just too small to see."

Don't trust the Cat in the Hat.

Follow up questions.

• How is it that all those cats fit under Cat in the Hat's hat?
• If all the cats stood on top of each other, how tall would they be?
• What would the ratio of successive cats have to be in order for Cat Z to be too small to see?
• How heavy are all these cats?
• What is voom and how does it clean up pink snow?