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 8.9×10^{-12} meters. Could you see this? The most common thing to compare small visible things is hair. Hair can be around 50×10^{-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?