Treebeard's Stumper Answer 10 October 2003

Alice Big and Small

Alice changes her size at least twelve times during her adventures in Wonderland. (Can you find and maybe graph all the changes?) Alice grows taller than the trees and as small as a mouse. It's a powerful reminder of the awkward changes of adolescence. I wonder what it would really be like to grow so much larger or smaller. Would everything stay the same except your size, or would there be unexpected consequences? (Hint: think about insects and mice and elephants!) What if everything suddenly doubled in size. Could we tell?

John Tenniel's classic illustrations of Alice normal, big, and small.

A fun bonus stumper is to find other books, movies, comics, etc. where ordinary people and/or things get really big or small. The fuzzy picture to the right is a frame from one of my favorite Tex Avery cartoons that really takes the genre over the top. Another cartoon favorite is the Simpson's episode where Itchy and Scratchy keep pulling out bigger and bigger handguns until they blow up the world! (I can't find a picture.) What else can you find?

I can imagine Alice growing larger than the trees in Wonderland, but just try to visualize a real elephant doing somersaults! Height is a single dimension, but area is size squared and volume is size cubed. If the real Alice tripled in size, her mass would increase 3³ = 27 times, but the cross-sectional area of the bones and muscles that support her would only increase 3² = 9 times. The real giant Alice would probably collapse with broken bones. If everything doubled in size (*ZAP*), I don't think we could tell. There's some dicussion below.

Notes:

I'm assuming that Alice keeps the same density when she triples in size. She is made of the same stuff, just 27 times more of it, volume and mass. Just standing up at that size would be like carrying two more of you piggyback. Even if you could, you wouldn't want to jump or fall!

Graybear agrees and points out another problem with heat:

The most notable thing about changing size is the differing ratios of length, area, and volume. In other words, if an object doubles in size (i.e. length), its area quadruples and its volume is now eight times the original. This can have at least two major impacts in humans, the first of which is heat transfer. At twice as much volume per unit of surface area, our sweat glands would work overtime to regulate our body temperature. The second impact is that ... weight is proportional to volume, thus placing twice as much stress on our bones and muscles. Between the two effects, only people in excellent physical conditon would be able to stand up. A 100-lb middle-schooler would now feel like they weigh 200-lbs, and if the ambient temperature were about 95 degrees, it would now feel like 120 degrees.

It's a matter of geometry. There is a difference between surface area and volume. Volume occupies space, how much stuff there is inside. Surface area is how much outside there is to a thing. Volume depends on size³ and area depends on size ², so the ratio of surface area to volume changes as size changes.

Consider a cube that is one unit on each side. It's volume is 1 x 1 x 1 = 1 cubic block. Each of its six sides has an area of 1 x 1 = 1 square block, so its total surface area is (1 x 1) x 6 sides = 6 square blocks. The ratio of surface area to volume is 6/1 = 6.
What happens to the area and volume if we double the size? The volume is now 2 x 2 x 2 = 8 cubic blocks. Each face has a surface area of 2 x 2 = 4 square blocks, so the total surface area is now (2 x 2) x 6 sides = 24 square blocks. The ratio of surface area to volume is now 24/8 = 3, exactly half what it was before.

The math is harder with an odd shape like Alice or you and me, but the result is the same. Doubling size cuts the surface-to-volume ratio in half every time. This has important consequences for living things at any scale. I've gained some weight over the past few years, so I know some of this firsthand!

Big critters have a relatively small surface-to-volume ratio.

• They carry proportionately more weight and lose less body heat.
• They need a food supply proportionate to surface area, so they don't have to eat as much.
• They pant or sweat to lose excess water.
• They need to actively move air inside, and they need elaborate circulatory systems to reach all cells.
• Heart and lungs have to supply more body mass, so they work harder. Big fish with gills have to keep moving, and even trees have problems supplying the canopy.
• They can live in colder climates because they hold their heat better.
• Keeping cool can be a bigger problem than keeping warm.
• They can take a water bath to cool off.
• Other large things like logs and boulders are probably obstacles.
• It's hard to do a pull-up, let alone a somersault.
• They have to worry about jumping and falling too far.
• Get too big and they have weird effects with the gravitational force.

Small critters have a relatively high surface-to-volume ratio.

• They carry proportionately less weight and lose more body heat.
• They need a food supply proportionate to surface area, so they have to eat more.
• They lose heat quickly, so they have to eat constantly to stay warm.
• They have little reserve, so they die if they lose water.
• Many small critters are "cold-blooded" and shut down when they can't stay warm.
• Tiny critters like planaria and insects don't need lungs, they just absorb oxygen through their skin by diffusion and it gets around.
• Getting wet can be dangerous because of the proportionately greater weight of the water they carry.
• Getting wet might be impossible because of surface tension, think about Water Skaters!
• Other small things like leaves and pebbles probably aren't obstacles.
• They can run up trees and even glass, and maybe hang from the smooth ceiling.
• They can jump far and move fast and run away after falls that would kill you and me.
• Get too small and they have weird effects with electrical and atomic forces.

The classic essay on this stumper is J.B.S. Haldane's "On Being the Right Size" (1928): "Comparative anatomy is largely the story of the struggle to increase surface in proportion to volume." His comment about falling 1000 yards down a mine shaft is more graphic: a mouse "gets a slight shock and walks away..., a rat is killed, a man is broken, a horse splashes." A real Alice growing big and small would face many unexpected and unpleasant consequences. The fantasy is more fun!

We're all reading Lewis Carroll's Alice in Wonderland at my school this year. Martin Gardner's Annotated Alice (2000) is a fun guide. He claims (p. 17) that Alice goes through twelve size changes in the book. Of course I set out to find and graph them all. Yellow is smaller, and blue is bigger.

Change Number Size Change Reference Where, Why, Size     Chapter 1, p. 11 Alice begins her adventure at normal size, say 3'6" or 42 inches, about right for an eight year old girl. 1 down Chapter 1, p. 17 At the bottom of the rabbit hole by the little door to the garden that is 15 inches high, Alice drinks from bottle that says "DRINK ME". Alice is now about 12 inches high. (See change 11.) 2 up Chapter 1, p. 18 At same place, Alice eats from the cake that says "EAT ME". Alice is "rather more than nine feet tall". 3 down Chapter 2, p. 24 At same place by the pool of tears, Alice shrinks because she is holding the White Rabbit's fan. She drops the fan "just in time to save herself from shrinking away altogether". Alice is about the size of a mouse, say four inches. 4 up Chapter 4, p. 38 In the White Rabbit's house, Alice drinks potion in an unlabeled bottle. She grows to fill the rabbit-sized house with "one arm out the window and one foot up the chimney". She is "a thousand times as large as the rabbit", if that's volume, then maybe 12 feet? 5 down Chapter 4, p. 44 Alice eats one of the pebbles that turn into little cakes, and shrinks again. She is mushroom size, three inches "is such a wretched height to be". 6 down Chapter 5, p. 53 Alice eats from one the right-hand side of the mushroom and shrinks so fast that "her chin ... struck her foot". She is less than one inch. 7 up Chapter 5, p. 54 Alice eats from the left-hand side of the mushroom and grows so tall that "all she could see, when she looked down, was an immense length of neck, which seemed to rise like a stalk out of a sea of green leaves that lay far below her." She is maybe 100 feet tall? 8 down Chapter 5, p. 56 Alice nibbles carefully from both sides of the mushroom to bring her size down to her usual height. She is now her normal 3'6" tall again. 9 down Chapter 5, p. 56 Alice nibbles still more from the right-hand side of the mushroom to bring her size down even more to enter little house where the Duchess and the Chesire-Cat live. She is now nine inches tall. 10 up Chapter 6, p. 67 At the March Hare's house, Alice nibbles more from the left-hand side of the mushroom to bring her size up for the Mad Tea Party. She is now about two feet tall. 11 down Chapter 7, p. 78 After the Mad Tea Party, Alice goes through a door in a tree and is back at the little door at the foot of the rabbit hole. Alice takes the key from the table and then nibbles more mushroom and shrinks to fit the door. She is now 12 inches tall. 12 up Chapter 11, p. 113 At the trial, the Hatter, a witness, takes a bite from his teacup. Alice feels herself start to grow but she stays in court. I'm not sure how large she grows, but the Red Queen tries to enforce "Rule Forty-two. All persons more than a mile high to leave the court." (p. 120) I choose to believe she's grown in maturity and has actually returned to her original height. She is again 3'6" or 42 inches.     Chapter 12, p. 125 The dream ends, and Alice is back to normal size. End of story.

Here's my graph. A linear graph over chapters would be nice, but that won't do in Wonderland. Alice mentions how she must "beat time when I learn music", and the Mad Hatter answers (chapter 7, p. 72) "[Time] won't stand beating. Now, if you only kept on good terms with him, he'd do almost anything you liked with the clock." I divided chapters into sections based on Alice's size. Time takes care of itself.

There are many books, movies, and comics that play with the rules of physics and biology to make ordinary things really big or small. This subject deserves it's own web page, but I don't have time. Here are a few I thought of before I started searching. I soon gave up because I found sites like GiantMonsterMovies.com that do it better. The first source I thought of was The Incredible Shrinking Man, both Richard Matheson's fine novel and the classic movie version. It would be fun to rank all these by size and physics!

The Incredible Shrinking Man 
Gulliver's Travels
David and Goliath
The Borrowers
Tom Thumb & Thumbelina
The Indian in the Cupboard
James Blish's "Surface Tension", part of "The Seedling Stars"
Steve Martin "Let's Get Small"
Jack and the Beanstalk
Anything by Ray Harryhausen!
Fantastic Voyage
Darby O'Gill and the Little People
The Incredible Shrinking Woman 
Honey, I Shrunk the Kids & Honey, I Blew Up the Kids
King Kong & Mighty Joe Young
Them!
The Fly
Simpson's Bible Stories & that Halloween episode where all the signs come alive
Tex Avery "King-Size Canary"
Plastic Man & Atom Man & Elastic Lad comics
Attack of the 50 Foot Woman
The Amazing Colossal Man
Godzilla, Mothra, Rodan, Gamera & all the rest from Monster Island 
Attack of the Giant Leeches
Attack of the Crab Monsters
Tarantula
The Deadly Mantis 
Earth Vs the Spider 
Bambi Meets Godzilla 
There's lot's more!

If Alice changes size and everything else stays the same, it would be easy to tell that something is different. But what if everything doubled in size? Mass and time stay the same, but every linear dimension doubles in size - rulers, atoms and electrons - everything! Sure, desities by the old rulers would change, but corks would still float. Could we tell that something is different?

I believe this stumper was originally posed by the mathematician Jules Henri Poincare (1854-1912) a hundred or more years ago, and it's still being debated. The question is whether all those squares and cubes cancel out or is there something we could detect even with changed rulers and physical constants, maybe down at the level of electron orbits and hydrogen bond distances.

I need more time to think about this. After 100 years of discussion, there's no hurry. (*ZAP*)

Here are some links for your own research:

• With the RIAA crackdown on file sharing, I don't have links to Steve Martin's "Let's Get Small" or The Jefferson Airplane's "Go Ask Alice" recordings. But I bet you can find them on the Web without getting in trouble! At least I can show lyrics:
  

  One pill makes you larger
  And one pill makes you small,
  And the ones that mother gives you
  Don't do anything at all.
  Go ask Alice
  When she's ten feet tall.
  And if you go chasing rabbits
  And you know you're going to fall,
  Tell 'em a hookah smoking caterpillar
  Has given you the call.
  Call Alice
  When she was just small.
  When the men on the chessboard
  Get up and tell you where to go
  And you've just had some kind of mushroom
  And your mind is moving low.
  Go ask Alice
  I think she'll know.
  When logic and proportion
  Have fallen sloppy dead,
  And the White Knight is talking backwards
  And the Red Queen's "off with her head!"
  Remember what the doormouse said:
  "Feed your head. Feed your head. Feed your head"
  

• Go Ask Alice! is also an important health question and answer Web service provided by Columbia University. All kids should know about this site for those sensitive questions they can't or won't ask at home.

• Lewis Carroll's Alice books are available here and here on the Web from Project Gutenberg. ClassicNotes has useful pages on Alice themes and a good chapter summary. Also see Shapes of Nature and Breaking the Looking-Glass.

• J.B.S. Haldane's "On Being the Right Size" (1928) is available on the Web. D'Arcy Thompson's "On Growth and Form" (1916) is another classic source that is back in print.

• The Big Cool Chicken discusses the real effects of raising bigger chickens. There is an excellent discussion of heat loss and surface/volume ratios with some math. Interesting site!

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