Dot Physics

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.

i-22451d08253937c2f9bda2e7329ec515-2010-03-01_cata_1.jpg

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):

i-fdd20b33ceb720bbd06a0b893d37251a-2010-03-01_cat_a_1.jpg

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

i-7010b80590304c7778f64342d6e9103f-2010-03-01_cat_b_1.jpg

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

i-d3af152106b84c201e6bfb479a703541-2010-03-01_cat_z_1.jpg

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.

i-d5fd8ee82fdbca36add24e0e56322d0e-2010-03-01_cat_a_b.jpg

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

i-1679b061ac5f7f1826f14cc371ad8f1b-2010-03-01_sequence_cat_a_b.jpg

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.

i-4172e3aec4a48c2ac9162c009fbf0beb-2010-03-01_cat_a_b_c.jpg

This gives:

i-005ce82c875f2efb0c996584b8e85485-2010-03-01_cat_abc_ratios.jpg

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.

i-642f1e18850d9af8c9724fe52851b567-2010-03-01_cats_cdefg.jpg

i-074fead9783ddd81bd976b95e25b028a-2010-03-01_cats_defg_ratios.jpg

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:

i-625c237f6b22dd5cb3c30654b25c021b-2010-03-01_cat_z_do_2.jpg

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

i-96151ef8f34f9a27d4721b48d6e64e33-2010-03-01_cat_z_hair_sized.jpg

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?

Comments

  1. #1 JimP
    April 28, 2009

    Dan Meyer, but you gave away the question!

  2. #2 Rhett
    April 28, 2009

    Dan Meyer is the Cat in the Hat! Should I not trust him?

  3. #3 Susan B.
    April 29, 2009

    Awesome analysis. Mind if I swipe this for the next calculus course I teach?

  4. #4 Rhett
    April 29, 2009

    Susan,

    If you like it, use it. That is what makes the internet great.

  5. #5 dr. dave
    April 30, 2009

    Next up: a Fermi problem to estimate how long it would take the Onceler to de-forest an entire Amazon Rainforest of truffula trees.

  6. #6 dr. dave
    April 30, 2009

    Anyway, looking at the pictures, it’s pretty clear that Mr. Geisel took some liberties with those drawings. Even crouching, a lot of those cats wouldn’t fit under the hat of the preceding cat. Thus, I think the first estimate is the better one, and cat Z would be something like nucleus sized. This would also explain the energy release of VOOM, which would then have to be based on nuclear rather than purely chemical processes.

  7. #7 theresa
    May 3, 2009

    waaaayyy to much time on your hands there bud! lol! but i had a good laugh.

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