Tuesday, January 5, 2016

Self-Consistent and Coexisting Spatial Scales in Spongebob Squarepants

Sometimes things initially don't make sense but upon closer investigation, they actually do make sense. 

This is the case with Spongebob Squarepants. If you this does not interest you, it would be best if you stopped reading now. I warned you.

For those of you who don't know, Spongebob Squarepants is a cartoon about a kitchen sponge and his wacky undersea adventures. He lives in a pineapple and works for a crab at a fast food restaurant in a treasure chest and does all sorts of other stuff too. Because its a cartoon for children, those kind of creative liberties are allowed. However, there is one critical aspect of Spongebob's universe that I found to be deeply unsettling. 
How can all of Spongebob's friends hang out when their sizes range over orders of magnitude? Pearl the whale is friends with Plankton the plankton and spongebob the sponge but most whales are at least 8 meters long, many plankton are about 20 micrometers, and kitchen sponges range from about 5-7 inches in length. How can this be?

After rigorous investigation of this seemingly inexplicable phenomenon, I have deduced that the Spongebob Squarepants universe exists on three self-consistent and coexisting spatial scales. 

There are three classes of articles in the herein purported multi-scalar meta-space where Spongebob takes place: characters, buildings, and non-building objects (NBO's). Each undergoes a different transformation from real space (R) to spongebob space (SS). 

Characters are the most interesting class. When you look at the sizes of the characters in R, it looks like this: 

But when those same characters in SS, their sizes look like this: 

Is there a mathematical function that we can use to transform a real-space object into spongebob space? Let's find out.

Here's what we start with:

We want to know exactly what to do to the size of each real object to make it the size that it should be in spongebob space. This means if we perform the correct operation on the size of an object in R and plot against the size of that object in SS, it should be completely linear.

Let's find the function that would achieve that goal.

Because those numbers vary so much in size, lets see what would happen if we take the log of the size of real objects:

Better. It looks like spongebob characters exist in a log-normalized version of real space! Cool. Now if we fit a linear equation to the log-normalized data we get this:

We could also use a higher order polynomial to fit the data a little more accurately like this:

I prefer the linear fit because while it isn't the most accurate, it has a consistent upward trend unlike the cubic fit. By applying the linear transformation to the log normalized data we make the residuals pretty small:

Hooray! A pretty good function for transforming an object from R into SS is:

With this helpful function, we can take any everyday object and approximate how big it would be in Spongebob assuming it follows the same rules as the other characters.

The best fit of the equations that transform buildings and non-building objects from R into SS are probably two linear functions. Some objects undergo one linear transformation to become buildings (Pineapple, Krusty Krab treasure chest etc.), while other objects undergo a different transformation to become non-building objects (Spongebob's pants, Squidward's clarinet, etc.) Maybe I'll figure out those at some point but at this point I've definitely wasted enough time on this 'project'.

I guess the only important thing that we learned here is that initially Spongebob space appears to be nonsensical and random but it is actually quite principled and logical! I definitely didn't cherry pick the characters that I used or the sizes that I assigned them. 
All done. Spongebob is completely reasonable. Problem solved. 

SpeciesSize (cm)SourceSize (su)Source
Kitchen Sponge12.7http://www.shop3m.com/3m-commercial-size-sponge-c41-7-5-in-x-4-375-in-x-2-06-in.html1http://www.amazon.com/Best-Sellers-Home-Kitchen-Household-Cleaning-Sponges/zgbs/home-garden/15754811
Whale396.24 -3048http://frontiersofzoology.blogspot.com/2012/02/whale-scale.html1.9http://spongebob.wikia.com/wiki/SpongeBob-Pearl_Relationship
Sea Snail<12http://www.snail-world.com/0.4http://spongebob.wikia.com/wiki/File:Patrick_SpongeBob_and_Gary_Opposite_Day.png

I need to stop drinking so much coffee.