### Differential Equations Final Review

Mar. 17th, 2014 10:34 pmThis evening I spent nearly 2.5 hours running the final review session for the differential equations students (their final is tomorrow morning). There were 46 students there! They ran out of chairs, so they sat on the floor; they ran out of floor, so they stood in the hallway. I gave them general tips on test-taking, I gave them specific tips on this particular professor’s tests, and I walked them through long and complicated problems on weighted strings, vibrating membranes, eigenvectors, and quantum oscillators. I told them that no matter what happens tomorrow, they’ve survived a full quarter of the toughest lower-division math class that this university offers, and that no matter what classes in math or physics or engineering they take from now on they’re always welcome to come to my office hours for help. When we finally called it quits so I could catch the bus I got a hearty round of applause and a bunch of handshakes. Good times and a satisfying end to the quarter.

"Can we make matrix snowflakes?"

"What? You mean like folded cutouts?"

"Yeah; punch holes in symmetric locations across the diagonal."

"I suppose. They'd be rectangular rather than hexagonal though."

"That's okay. Salt snowflakes!"

"I guess you might have salt snow on a planet with a reeeeeally high temperature."

"When it rains, it pours!"

"Ooh, nicely done!"

### 10^n Sour Patch Kids

Jan. 17th, 2014 07:40 pmhttp://waitbutwhy.com/2013/10/what-

### On Spirals, Seashells, and Gullibility

Mar. 9th, 2013 02:00 pmHOWEVER.

The Fibonacci Spiral and the Nautilus, or, don't believe everything you read

Sea Shell Spirals

The Myth That Will Not Go Away

The final link sums it up: "The Nautilus does grow its shell in a fashion that follows a logarithmic spiral, i.e., spiral that turns by a constant angle along its entire length, making it everywhere self-similar. But that constant angle is not the golden ratio."

In other words, just because something is a spiral doesn't mean it's a "golden spiral." The golden ratio does have some very interesting mathematical properties, but anything anybody tells you about its relation to aesthetics or architecture or seashells or flowers or galaxies or human anatomy is almost certainly false.

From what I've read, it sounds like the whole idea that the golden ratio shows up

*everywhere*was invented and promoted by some 19th century architects who believed that the number had some sort of mystical significance.

Ultimately the blame lies with human gullibility. When somebody says something like "Did you know that the distance from your feet to your navel and the distance from your feet to your head form the golden ratio?" the usual response is "Wow, that's neat!"

THIS IS A SERIOUS BUG IN HUMAN PROGRAMMING.

The correct response is "Interesting; let's get a measuring tape and a bunch of people and see if that's really true!"

### Physics workshop, coffee!, and derivatives

Feb. 1st, 2013 12:53 amToday in my physics workshop I taught about some water flow stuff, some electrical flow stuff, a lot of heat flow stuff, and an introduction to an equation that ties all of them together. I love covering topics like this because much of it is new ideas that the students are seeing for the first time, yet can be related directly to their everyday experiences.

In particular, I was discussing the idea of thermal conductivity--a measurement of how easily heat can pass through a material. "When would you want low conductivity?" I ask the class; "what's a situation where you want heat to flow very slowly?"

Silence. Somebody ventures "Chemical reactions?"

"Stop thinking about the lab for a minute," I say. "What's an *everyday* thing that you want to stay hot for a long time?"

Thoughtful silence for a second or two. Then, all at once, half the students in the room say "…Coffee!"

(Why yes, it is midterm season; how did you know?)

So this leads into a great discussion of the sorts of materials that are used for coffee containers and why they work well and others don't. Good times.

Even better: the heat transfer stuff lead to some examples with exponential decay, and investigation about why the energy vs time graph follows exponential patterns. After the workshop, two of the students asked for some clarification about exponential growth and decay--why does it do that? Why is the number "e" so important? It's not like the bacteria know about "e," do they? So I went into my usual explanation of what the derivative really means, how it relates to the basic prealgebra definition of slope, why it all works, how that applies to exponential functions, and why we use "e" as the base (short answer: it makes derivatives easier). The students just ate it up. One of them even showed up to my office hours later, asking if I could go over the meaning of a derivative again so she could think about it some more and make sure she's got it.

It took a while for that to really sink in: two students who have already been through calculus, and never need to take a math class again, voluntarily stuck around after an optional workshop because they wanted me to explain derivatives.

Sometimes I forget that my teaching skill really has improved quite a bit since I first started seven or eight years ago. It's nice to get a reminder of that every once in a while.

In particular, I was discussing the idea of thermal conductivity--a measurement of how easily heat can pass through a material. "When would you want low conductivity?" I ask the class; "what's a situation where you want heat to flow very slowly?"

Silence. Somebody ventures "Chemical reactions?"

"Stop thinking about the lab for a minute," I say. "What's an *everyday* thing that you want to stay hot for a long time?"

Thoughtful silence for a second or two. Then, all at once, half the students in the room say "…Coffee!"

(Why yes, it is midterm season; how did you know?)

So this leads into a great discussion of the sorts of materials that are used for coffee containers and why they work well and others don't. Good times.

Even better: the heat transfer stuff lead to some examples with exponential decay, and investigation about why the energy vs time graph follows exponential patterns. After the workshop, two of the students asked for some clarification about exponential growth and decay--why does it do that? Why is the number "e" so important? It's not like the bacteria know about "e," do they? So I went into my usual explanation of what the derivative really means, how it relates to the basic prealgebra definition of slope, why it all works, how that applies to exponential functions, and why we use "e" as the base (short answer: it makes derivatives easier). The students just ate it up. One of them even showed up to my office hours later, asking if I could go over the meaning of a derivative again so she could think about it some more and make sure she's got it.

It took a while for that to really sink in: two students who have already been through calculus, and never need to take a math class again, voluntarily stuck around after an optional workshop because they wanted me to explain derivatives.

Sometimes I forget that my teaching skill really has improved quite a bit since I first started seven or eight years ago. It's nice to get a reminder of that every once in a while.

- Current Mood: accomplished

### Multiples of relative primes

Dec. 20th, 2012 09:50 pmCan anyone find a counterexample? Or any thoughts on a proof?

...

Possible generalization: for any number n that is relatively prime to B, there is a multiple of n that is equal to 1+B+B^2+B^3+...B^x for some x.

### Lego gearbox

Oct. 15th, 2012 12:30 am(The stack on the right contains a 9V battery in the upper compartment and a bank of capacitors in the lower compartment. The switch allows toggling between charging and discharging the capacitor bank through a buzzer, demonstrating exponential decay of current.)

On a whim, I decided to get some axles and gears as well. Between workshops and office hours, I occasionally spend some time in my office messing around with these delightful clockwork building blocks, and this is what I've managed to build:

The cream-colored gears have 20 teeth each and the black ones have 12, so each axle rotates at a speed 20/12 times that of the previous one. That means the final axle rotates at (20/12)^6 times the speed of the first (hey! more exponential growth!), or about 21 times as fast. I've ordered a few more gears and axles (I ran out) so I can fill up the rest of the block and get a 60x speed multiplier!

See it in action here:

### Programs on my graphing calculator

Oct. 9th, 2012 12:10 pmAAA: a placeholder for small temporary programs, e.g. if I want to experiment with some sequence to look for patterns but don't need to keep the coding for later.

ABCSPRAC: Abacus Practice; generates two random numbers and an operation (+, -, x, ÷), waits for user to press Enter, and displays the answer.

ANT: checks if current pixel is black or white; if black, changes it to white, turns left, and moves forward; if white, changes it to black, turns right, and moves forward. Leads to some interesting designs, especially if there's already something on the screen. Based loosely on a discussion of emergent properties in

*Science of Discworld*.

BAB3: uses the Babylonian Method to approximate cube roots (see below).

BABYLON: uses the Babylonian Method to approximate square roots. Suppose you want to find the square root of 12. Start by picking a guess; let's say 3. But 3*3 isn't 12; 3*4 is 12. This means that the square root of 12 must be between 3 and 4. The midpoint would be 3.5, so use 3.5 as a new guess and start the process over. But 3.5*3.5 isn't 12; 3.5*3.429 (approx.) is 12, so the square root of 12 must be between 3.5 and 3.429. The midpoint would be 3.4645, so we use 3.4645 as a new guess and start over... and so on. By hand it's tedious, but a calculator program can perform such repetitive tasks extremely quickly. I wrote this for a class in grad school called Topics in Analysis.

BALL: displays a single pixel bouncing off the walls of the screen (and tracing its path). Initial velocity is chosen randomly; velocity can be influenced using the arrow keys.

BALLGRAV: same, but with gravity.

BIRTHDAY: calculates the probability that at least two people have the same birthday in a group of

*n*people.

BISECTIN: the Bisection Method; approximates the x-intercept of a function in a similar fashion to the Babylonian Method. The user picks two numbers as guesses (an upper bound and a lower bound between which the intercept must be); the program finds the midpoint between those guesses and uses that as either the new upper bound or the new lower bound (depending on whether the function's value there is positive or negative) and starts the process over again. Also for Topics in Analysis.

BOUNCE: displays the word "BOUNCE" bouncing around the screen. Rather silly, I'm afraid.

BRIKDRAW: an attempt to make a Breakout-style game. It kinda works but it's really boring and there's no way to lose.

BURNSHIP: an attempt to draw the Burning Ship, a fractal somewhat similar to the Mandelbrot Set. Way too slow. I should try it in LOGO instead.

CIRCINV: displays a circle on the screen, allows user to select a point, and draws the circular inversion of that point (sort of like a reflection, but across the circle rather than across a line). It's useful in certain branches of geometry and I think it might be related to cylindrical mirror anamorphoses. M.C. Escher was probably into it too.

CTROFMAS: given two point masses and the distance between them, finds their center of mass. I wrote it for my own use when I was teaching high school algebra/physics as a quick way of checking students' answers.

DICE: prompts the user for number of dice and number of sides per die, then simulates rolling those dice and displays the total. A nice time-saver for RPGs.

DIRFIELD: given a differential equation, draws a direction field for it. The solution to a differential equation is a curve, or rather a family of curves; the direction field is a grid of small line segments whose direction shows which way a solution curve would be going

*if*it passed through that point. A very useful visualization tool.

DRAWPTRN: fills in pixels on the screen based on a certain algorithm to see if interesting patterns result. Currently I have it set to "if the x-coordinate and y-coordinate are relatively prime, color the pixel black," which does indeed yield an interesting pattern. Might look nice on fabric. I am reminded of the bit in

*The Difference Engine*about a fabric made by Ada Lovelace "feeding raw algebra into a Jacquard Loom."

DRAWRAND: same, but the algorithm is "flip a coin to decide whether to color the pixel black or white." Silly.

E: displays the values of (1+1/n)^n as n grows larger and larger, demonstrating that this gradually approaches 2.71828... or

*e*.

EULER: uses Euler's Method to approximate the solution of a differential equation. It's essentially a numerical version of the direction field mentioned above.

EULERPAR: same, but for parametric equations.

FACTOR: given a whole number, lists all pairs of factors whose product is that number (e.g. given 45, it will list 1, 45, 3, 15, 5, 9).

FIBONACI: lists out as many Fibonacci numbers as the user requests.

GCDEUCLID: uses Euclid's Method to find the greatest common divisor of any two whole numbers.

GR8RACE: asks the user for the strength of gravity, the mass of the race car, the length of the race track, the coefficient of friction, the force the engine can provide, and the car's desired initial and final velocity, and tells the user where the driver should hit the brakes. I wrote this as a quick way to check students' answers for a project in the algebra/physics class.

JACOBI: I have no idea what this does. It appears to be an iterative method for approximating the solution to some sort of matrix equation, so I presume it was for Topics in Analysis. Probably something like the Bisection Method but with matrices?

LAUNCHV: given two numbers A and B and the strength of gravity, tells the user the initial speed and direction with which a projectile must be launched. I'm not quite sure what A and B represent; possibly the coefficients in the equation of the parabolic path the projectile is intended to follow. Another one for checking students' work in the algebra/physics class.

M47R1X: gradually fills the screen with random ones and zeroes. Silly.

MANDLBRT: draws the Mandelbrot Set by making a sequence of calculations for each pixel on the screen. Works well but is very very slow. I'm proud of it but I like my LOGO version better. Amusing trivia: I used the chorus of JoCo's song "Mandelbrot Set" song as my sole reference for programming this.

MANDLCHK: originally just checked if a single point, given by the user, is part of the Mandelbrot Set or not. Useful for checking students' work in the algebra/physics class. I later modified it to also display some neat graphs showing how exactly the aforementioned sequence of calculations converges.

NEWTON: uses Newton's Method to approximate the x-intercept of a function. It starts with a user's guess as usual, then treats the function as a straight line near that guess and figures out where that line would hit the x-axis (easy enough) and uses that x-value as a new guess to start the process over again. Very effective. Another one from Topics in Analysis.

ORANGE: did you ever see that old logic puzzle where you start with a cup of water and a cup of orange juice and you put a spoonful from cup A into cup B and stir, then put a spoonful from cup B into cup A and stir, and so on over and over again? No? Well anyway this program simulates that.

PASCAL: draws Pascal's Triangle with all multiples of

*n*removed. Very neat patterns, especially for prime

*n*. I wrote this while student-teaching Algebra II (and later used it in a presentation for Topics in Analysis) and I'm very proud of it.

PASCLROW: displays all the numbers in row

*n*of Pascal's Triangle.

PASCNORM: displays some scatterplots that are probably intended to imply that the larger rows of Pascal's Triangle begin to approximate the normal distribution from stats? I'm not sure I remember writing this one, but it's definitely my style.

PASSWORD: this isn't a program; it's just where I store the password to the current level on a built-in puzzle game called Blockman. I haven't actually played it in years.

PCOUNT: not quite sure what this does. I think I was using it to count how many steps it takes for a certain p-series, specifically (10/11)^1 + (10/11)^2 + (10/11)^3 + ..., to reach 9, then 9.9, then 9.99, then 9.999, then 9.9999, etc. as it converges towards 10. I think I felt like I was on the verge of discovering something important but then realized that, when examined from a different point of view, is was pretty obvious and mundane so I gave up. I forget what exactly it was though.

PRFACTOR: similar to FACTOR, but instead of listing pairs of factors, it breaks the number down into its prime factors. I was rather proud of this one; it's not a very efficient algorithm but it's one I made up myself and it's a little faster than brute force.

PSERIES: similar to PCOUNT, but without a counter to keep track of how many steps it's taken.

PTRSBURG: simulates the so-called "St. Petersburg Lottery," a somewhat famous exercise in probability theory about a hypothetical gambling game set up in such a way that the only way it can be truly fair is if the cost to play is infinite.

PUNKED: another prank. When the program is running, it

*looks*like the ordinary home screen, and allows the user to type in most simple calculations as normal, but multiplies each result by a random modifier between .99 and 1.01... leading to numbers

*close*to the answer but with something

*wrong*about them. Intended for use on students who rely too much on their calculators but never actually deployed.

PYRAMID: an attempt to extend Pascal's Triangle into three dimensions. Enter a number and the program will give you a square matrix representing a cross-section of Pascal's (Square) Pyramid that many layers down from the top. I've since realized that a pyramid whose cross-sections are equilateral triangles is actually more useful, but this one still has some neat patterns.

QUADFORM: uses the quadratic formula to solve a quadratic. Occasionally useful; mainly I keep it around as a quick and simple example to show students how programming works using a mathematical concept they already know.

QUADITER: uses an iterative method similar to the Babylonian Method to approximate the solutions of a quadratic--as usual, user picks a guess and program repeats some calculation to get a closer guess, and a closer guess, and a closer guess, and so on. Written for Topics in Analysis, of course.

ROCKET: given three numbers (acceleration of gravity, rocket's velocity when observed, and height at which it was observed, I think?), the program calculates when the rocket was launched, when the rocket will land, and how high it reaches. Another one for the physics/algebra class, of course.

ROCKET2: a prank on my co-teacher. I described it to him as "an improved version of the ROCKET program," but it exaggerates the launch time by a factor of 100, gives the land time as π seconds, and claims that the rocket will reach a height of "a shade of green" meters at time "fish" seconds. Silly.

ROT13: given a letter, converts to a number (A=1, B=2, C=3, etc.), then adds 13 and converts it back to a letter. A simple but classic way of encrypting a message. Fun fact: the process of decrypting this particular code is exactly the same as the process of encrypting it, because 13 is exactly half of 26.

SFBDICE: similar to DICE, but modified for use with the damage allocation chart in Star Fleet Battles.

SFBDSTNC: an attempt to create a program that could quickly calculate the distance between any two hexes on a hexgrid based on their coordinates using the hex-labeling system in Star Fleet Battles. Never figured out how to get it to work.

SFBSPEED: type in your ship's speed and this program will tell you when it should move during the turn. Very useful.

SIERPRND: uses a random-walk algorithm to generate, rather surprisingly, the Sierpinski Triangle. Fun to watch.

SLIDESHO: a quick way to display saved images without having to go through a bunch of menus each time. Nice shortcut for showing off related graphs, variations on Pascal's Triangle, zoom-ins of the Mandelbrot Set, etc.

SNOCRASH: creates some small files and makes them bigger and bigger until the calculator's memory is completely full. Silly.

STUDENTS: displays the name of a student, randomly chosen from a list. I used this when I was teaching high school to call on students randomly.

TEMP: converts temperature between celsius and fahrenheit. Another one I use to teach students the basics of programming.

TWENTY4: simulates an old French dice game called Twenty-Four (roll two dice 24 times; if you get double sixes at least once, you win; if not, you lose). Mainly of historical interest; apparently Pascal was inspired to invent modern probability theory when a friend asked him if this was a fair game or not. (Turns out that losing is more likely than winning, but only very slightly.)

TWENTY42: same, but runs the game many times and then displays how many wins vs. how many losses.

VECSPIN: user enters a 2x2 transformation matrix. Program simulates a vector of length 1 spinning around the origin, but displays the result of said vector multiplied by the transformation matrix instead.

VECTRAIL: same as above, but leaves a trail.

It's good to be back in the driver's seat of the Turtle.

You can run it here to see what it does but you'll want to use this modified version instead to fit their screen:

cs pu lt 90 fd 240 rt 90 pd make "x 3 repeat 9 [ repeat :x [ fd 20 rt 360 / :x ] pu rt 90 fd 7 * :x lt 90 pd make "x :x + 1]

### This is why we can't have nice units

Aug. 10th, 2012 05:21 pm"Why did you bop the back of my head?"

"Because you aren't keeping track of units!"

"But I'm converting kilograms to pounds!"

"That measurement's not in kilograms; it's in dollars per kilogram. If you had typed that in to the converter, you would have ended up with dollar-pounds per kilogram squared!"

(in case anyone's wondering)

I have mentioned in the past that division and multiplication are really the same thing, and addition and subtraction are really the same thing. What I mean by that is that any subtraction can be replaced by addition of the opposite, and any division can be replaced by multiplication by the reciprocal.

For example,

30 - 5

really means

30 + -5,

because -5 is the opposite of 5; that is, -5 is what can be added to 5 to cancel it out.

Likewise,

30 ÷ 5

really means

30 • 0.2,

because 0.2 is the reciprocal of 5; that is, 0.2 is what can be multiplied by 5 to cancel it out.

So if we say "divide by x," what we really mean is "multiply by the reciprocal of x."

Thus if we say "it is possible to divide by zero," what we are saying is "it is possible to multiply by the reciprocal of zero," or, in other words, "zero has a reciprocal."

Suppose for a moment that zero DOES have a reciprocal. Let's call it Z. If Z is really the reciprocal of 0, that means that

0 • Z = 1,

because that's the definition of "reciprocal."

Now consider the equation

5 • 0 = 0.

If Z really exists, we should be able to multiply both sides of the equation by Z.

5 • 0 • Z = 0 • Z.

But 0 • Z is 1, so we can substitute:

5 • 1 = 1,

or in other words

5 = 1.

So if you're willing to allow zero to have a reciprocal, you must also accept that 5 = 1, and indeed that any number is equal to 1 (there's no reason we had to use 5).

Clearly if we want math to make any sense whatsoever we can't have every number equal every other number. In order to avoid this, we must assume that zero has no reciprocal, and thus that division by zero is impossible.

### Closed-path temperature

Jun. 25th, 2012 03:32 pmSuppose you draw out a loop path on a map. Any map you want; any shape path you care to draw, as long as it starts and ends at the same point.

Is it

*possible*that every location along the path you drew is a different temperature? Or

*must*there be at least two locations on that path that are the same temperature? Why?

(Clarification: when I say "location on the path" I mean locations in the actual place represented by the map--but that doesn't actually matter; if we were instead discussing the temperature of different parts of the map itself, it would still be fundamentally the same question.)

This afternoon was a nice reminder of why I love teaching differential equations (and teaching in general). I was explaining a new method of solving higher-order differential equations with matrix algebra, using as an example a simple second-order differential equation similar to ones the students had solved by easier methods a few chapters earlier.

After a lot of math I finally got to a quadratic that would yield two critical numbers (the eigenvalues) that would become part of the solution, and asked the students "Where have you seen this quadratic before?" then pointed to the original problem and asked "How would you have solved that problem back in Chapter 3?"

"OH!" they said, "It's the characteristic quadratic!"

In other words, an important intermediary step in this more complicated but more generalized method ends up leading to the exact same equation (but for different reasons) as the simpler but more limited method they learned a few weeks ago. And the topic of differential equations is FULL of potential "aha!" moments like this!

And then after the workshop was over one of the students said "Thanks, Barnabas; you make math fun!" which was nice.

After a lot of math I finally got to a quadratic that would yield two critical numbers (the eigenvalues) that would become part of the solution, and asked the students "Where have you seen this quadratic before?" then pointed to the original problem and asked "How would you have solved that problem back in Chapter 3?"

"OH!" they said, "It's the characteristic quadratic!"

In other words, an important intermediary step in this more complicated but more generalized method ends up leading to the exact same equation (but for different reasons) as the simpler but more limited method they learned a few weeks ago. And the topic of differential equations is FULL of potential "aha!" moments like this!

And then after the workshop was over one of the students said "Thanks, Barnabas; you make math fun!" which was nice.

- Current Mood: accomplished

### Good day today!

Feb. 6th, 2012 05:51 pm### Entropy: Have you tried using logarithms?

Nov. 30th, 2011 01:01 amThis morning I was reading about entropy in preparation for teaching a physics workshop on the topic. It's been a long time since I last studied entropy and I didn't understand it very well in the first place, so I started with the basics.

ΔS = C · ln( T

said the textbook.

"Why exactly would there be a logarithm in the definition?" I asked.

Also, ΔS = Q / T, and don't forget that Q = C · ΔT,

said the textbook.

"Oh! Of course that leads to a logarithm," I exclaimed joyously.

This is one thing about the Barnabas of today that I really really like. The Barnabas of ten years ago (or even five years ago) would not have had that realization.

ΔS = C · ln( T

_{f}/ T_{i}),said the textbook.

"Why exactly would there be a logarithm in the definition?" I asked.

Also, ΔS = Q / T, and don't forget that Q = C · ΔT,

said the textbook.

"Oh! Of course that leads to a logarithm," I exclaimed joyously.

This is one thing about the Barnabas of today that I really really like. The Barnabas of ten years ago (or even five years ago) would not have had that realization.

- Current Mood: brilliant

### That's a lot of pennies

Nov. 22nd, 2011 10:05 pmBest of all though were the sincere thank-you-so-much-I-finally-understand from several students afterwards.

### THE SQUARE-CUBE LAW: An Explanation

Nov. 13th, 2011 01:06 pmNow consider surface area. The area of each face of the small cube is 4m^2 (2m*2m), and its total surface area is 24m^2. The area of each face of the large cube is 400m^2 (20m*20m), and its total surface area is 2400m^2.

Note that any area of the large cube is thus 100 times the corresponding area of the small cube.

Moving on to volume: the volume of the small cube is 8m^3 (2m*2m*2m), but the volume of the large cube is 8000m^3 (20m*20m*20m).

Thus the volume of the large cube is 1000 times the volume of the small cube. If they have the same density, the large cube is also 1000 times as massive as the small cube!

This relationship can be generalized to any two objects of similar shape but different size. Suppose the ratio of lengths is R; that is, the larger object is R times as tall (and R times as wide and R times as long, etc.) as the smaller object.

Then the larger object will have R^2 times the area of the smaller object, and R^3 times the volume (and mass, if same density) of the smaller object. (Hence "the square-cube law.")

And here's how this relates to strength:

The strength of a muscle (or any fiber, really) is directly related to its cross-sectional area. Thus as a creature grows in size (length) by a factor of R, its strength grows by a factor of R^2. However, its volume--and thus its mass!--grows by a factor of R^3.

For instance, suppose I were to grow to giant size, 10 times as tall as I am now. All other things being equal, I would be 100 times as strong! BUT I would also be 1000 times as heavy. Since my weight increased 10 times more than my strength, it would feel to me as if it took 10 times more effort just to hold up me own weight; I would feel only 1/10 as strong as before! This is one of several related reason why there is a practical limit to the size of land animals: a big enough creature wouldn't even be able to stand up. (In the sea, that concern is negated by buoyancy, but there are other problems: the heart, for example, must be strong enough [R^2] to pump blood throughout the body [R^3].)

Alternatively, suppose I were to shrink to the size of an ant--let's say 1/400 of my current height to make the calculations simpler. I would be 1/160000 as strong, but 1/64000000 as heavy, and 1/160000 is 400 times as big as 1/64000000, so from my perspective, I would be 400 times as strong!

So what does this mean? An important realization to get out of this is that the notion of super-strong little arthropods is not a biological issue, but a direct consequence of geometry itself. And no, basic geometry cannot be violated by a radioactive spider bite. :-)

If you find this interesting, you ought to read Haldane's "On Being the Right Size."