How Big Is Infinity? Even Bigger Than the Internet

... which is so big it feels infinite sometimes. I've been following hundreds of math-related blogs for the past few years. How delightful to discover a whole niche of math-related goodies I'd never dreamed of.

Have you heard of 3-D printers? They're a basic part of the plot of Makers, by Cory Doctorow, a sci-fi novel I read a few weeks ago. (I bought the UK edition at my local independent bookstore in California. You can download the book for free at Cory's website.) Those 3-D printers are already real. (I think I knew that, but I haven't seen one in action yet.)

Christian Perfect, over at Aperiodical, has just posted a roundup of cool mathy stuff being made on 3-D printers. You can get designs at Thingiverse if you have access to a 3-D printer (which cost between $700 and thousands of dollars), and you can buy stuff at Shapeways. Christian reported the prices in pounds, so I had to visit the site to see if things were affordable in dollars. Yep. If you'd like to give math-inspired earrings as a gift, you can do it. Or weird dice. Or beautiful little sculptures. (My math-loving niece is getting this one.)

I have to wonder whether Cory had Shapeways in mind as he wrote Makers. This line from Shapeways' webpage sounds like something right out of the book:

"Shapeways is a spin-out of the lifestyle incubator of Royal Philips Electronics. Investors include Lux Capital and Union Square Ventures in New York and Index Ventures in London."

The San Francisco Chronicle has an article on 3-D printers that might help you get oriented. I just spent an hour or so looking for articles and watching youtube videos. I didn't find much that's well-written or exciting to watch, but I have a better basic understanding now of what they are.

After I was almost done ordering those earrings, I saw that they'd ship after the holidays. Apparently some things will ship faster. It depends on the materials. Also, if you do buy earrings, make sure you're getting two; some are priced per earring.

Repost: Holiday Logic Puzzles

This is reposted for your puzzling pleasure from my original post 3 years ago.

The first puzzle shouldn't be too hard. The second ... is tough. It's by Mike Shenk, who has a site called puzzability, and is interviewed here. 

I wish you all peace and joy during these winter holidays.

Holiday Logic 
by Sue VanHattum

1. The Green girl’s favorite Christmas tradition is singing carols.
2. The Brown boy celebrates Kwanzaa indoors.
3. DJ and Jordon joined their friend in her candle lighting ceremony.
4. Layla and Amani joined their friend for his annual walk through the woods.
5. The Gold girl came to Jordon’s house to join his family in their feast.
6. The Fox family celebrate the Yuletide, and Amani comes to their party.
7. Amani couldn’t make it to the Gold family’s Hanukkah celebration.

Each child celebrates just one holiday with one special activity as a tradition in their family, though they do join in the fun with their friends this year. Your mission: Decide who celebrates each holiday, and what they do to celebrate.

Oh Deer! A logic problem by Mike Shenk
(first published in Games Magazine, December 1992)

Twas the night before Christmas, and at the North Pole
The last-minute planning was taking its toll.
As Santa was hastily making a scheme
For the placement of deer in his sleigh-pulling team,
The good Mrs. Claus was crocheting bright bows
To be worn by these reindeer (four bucks and four does).

The ribbons were colored in eight festive hues:
One ocher, one rose, one cerise, one chartreuse,
One maroon, one magenta, one white, and one blue.
(These ribbons helped Santa keep track of who's who.)
The deer pulled the toy-laden sleigh in four rows,
Arranged so no row held two bucks or does.

The order of pullers was changed year by year,
For Santa was thoroughly fair with his deer.
He summoned the elves and instructed them thus:
"Let's hitch up the reindeer with minimum fuss.
The bow on the buck behind Dasher is white,
While Blitzen, a doe, sees cerise to her right.

The blue bow is nearer my sleigh than is Dancer,
But nearer the front of my team than is Prancer.
The doe in chartreuse gets a front-of-team honor,
But not on the same side as Cupid or Donner.
Now Comet stands two spots ahead of the rose.
And three deer of four on the right side are does.

The cerise bow is worn two in back of maroon,
One of which is beside the bright ocher festoon.
Oh-Cupid's in front of a buck, by the way.
Well, that's how they line up for pulling my sleigh.
I trust that you elves, being clever, now know
Each reindeer's position and color of bow."

In no time each colorful ribbon was tied
And the team was hitched up for the transglobal ride.
Can you ascertain where each member fits in?
Who's Comet? Who's Cupid" Where's Donner? And Blitzen?
Who's Dasher? Who's Dancer" Where's Vixen? And Prancer?
With logical thought, you'll determine the answer
And write down the color and place for each deer.
Happy Christmas to all, and to all much good cheer!

Math Girls 2 and Differentiating

Last year, I posted a review of Math Girls, by Hiroshi Yuki. I just discovered that Math Girls 2: Fermat's Last Theorem is coming out on December 12th. The first two chapters are available online, and look just as good as the first book. I love Pythagorean Triples, and that's the topic of the second chapter.

I also wanted to mention the cool math teacher in the book. I like how he offers his best students some personal challenges.

Mr. Muraki was our math teacher. He had taken a liking to us, and would regularly slip us index cards with all sorts of interesting math problems. They rarely had anything to do with our classwork, which made for a refreshing change of pace. We always looked forward to what he would come up with next.

I like this. I think I'll try to do it next semester.



I'll write a more complete review once I've managed to read the whole thing.

Good Calculus Textbook?

My department will be looking for a new calculus textbook over the next few months. We used to use Stewart, but there was some discontent, and we switched in part due to the high price. We've been using Briggs for about two years, and are very unhappy with it. So we want to switch again.

I asked a month and a half ago for a good discrete math text, and Josh suggested Discrete Math With Ducks. It looks fun, and was inexpensive compared to what we had been using. I'm excited that my request for suggestions panned out. (Thanks, Josh!) I'll have more to say about that next semester when I start teaching from it.

That was a decision I got to make on my own. The calculus textbook will be a group decision. The rest of the department will want a more conventional textbook than what I might want. I'm willing to work with whatever textbook we use, but I'm dreaming now of writing my own. (That will take a few years...)

What we didn't like in Briggs:

• The exercises often jumped too quickly to very hard problems
• There's nothing on centroids (until multivariable)

Hmm, I know there's more - I'll have to add to that list next week after our department meeting. I'd like to bring suggestions to the meeting, though. Have any of you used a calculus textbook that you love? Do any of you know of a complete textbook (for Calc I, II, and III, ie going through multivariable calculus) that's under $100?

My department doesn't seem interested in open source textbooks, and the two I used this semester weren't impressive enough for me to want to push it. I love the projects in Boelkins, but that only works if you want to teach through projects. The Guichard made some odd choices. I think any open source book will have more of its own personality than the commercial books. That could be fine, but I haven't seen one yet that will cover all the bases for us.

Ideas?

Centroid (Center of Mass)

This is a topic covered in Calculus II. The textbook explanation is inadequate, and I found nothing good online. So I wrote my own explanation. I now understand it better than I ever did before. (Not surprising, huh? If you're a student, this is an important principle of learning. After you think you understand something, try to write an explanation of it and see how much deeper your understanding can get.)

I'd love to improve this, so please let me know where it's unclear.


Imagine a thin sheet of metal cut in an artistic shape. Is there always a spot where you could hold it balanced on your finger? If there is, can we find that spot? We’ll assume the metal has uniform density. This allows us to treat area as equivalent to weight.

The 1-dimensional Case
To think about this, we first imagine a teeter-totter. We know that two people of the same weight must sit the same distance from the fulcrum (balance point) to balance. We also know that a heavier person must move inward if they want to balance with a lighter person. Experiments show that the weights times the distances from the fulcrum must be equal on the two sides for the teeter-totter to balance. (I wonder if there’s a thought experiment we could do that would convince us this must be true, without the actual experimental evidence.)

Example 1: I weigh 170 pounds and sit 5 feet from the center. My son weighs 75 pounds and sits in front of me, 4 feet from the center. Weights times distances = 170*5 + 75*4 = 1150 feet-pounds. We need someone who weighs 230 pounds to sit 5 feet from the center on the other side. We could write this as:  w₁*d₁ + w₂*d₂ = w₃*d₃

If we change our perspective to a number line below the teeter-totter, with 0 at the fulcrum, then the values on the left will be negative. We won’t have the same equality – we’ll have

 w₁*p₁ + w₂*p₂ = -(w₃*p₃), 

where each d (for distance, always positive) was replaced with a p (for position). This becomes

 w₁*p₁ + w₂*p₂ + w₃*p₃ = 0, 

given that the fulcrum is at 0. But suppose we don’t know where the fulcrum is? Let’s just put our 0 at the left end, and let the former proper place for the 0 - at the fulcrum - be f. Then the equation becomes

 w₁*(p₁ - f) + w₂*(p₂ - f) + w₃*(p₃ - f) = 0, 

or  w₁*p₁ + w₂*p₂ + w₃*p₃ =  w₁*f + w₂*f + w₃*f, 

or w₁*p₁ + w₂*p₂ + w₃*p₃ =  f(w₁ + w₂ + w₃). 

Let W = the sum of all the weights, then we have

f = (w₁*p₁ + w₂*p₂ + w₃*p₃)/W,

which of course extends from 3 weights and positions to n weights and positions.

On to 2 Dimensions
If we use areas instead of weights, we can look for the fulcrum of the x-values (written as an x with a bar over it) and the fulcrum of the y-values (written as a y with a bar over it). For a finite number of small areas, we would get (the same as above)

xbar = (a₁*x₁ + a₂*x₂ + ... + a_n*x_n)/A. 

If we imagine a shape formed by the area under a function f (where f has positive y values), between x=a and x=b, sliced into infinitely many vertical strips, with the area of each vertical strip given by height times width = f(x)^. Δ x, then taking the limit as Δ x goes to 0 gives us

For the y value, we need to notice that the vertical center of each vertical strip is at 1/2 *f(x), and we use this instead of x for the position. So we get

 or

Many times the area we're interested in will not be touching the x-axis, and so we need area between a top function, f(x), and a bottom function, g(x). The height of each slice will be (f (x)- g(x)), making the area  (f(x) - g(x))^. Δ x. The vertical center is now given by averaging f and g. We get:

 

 or

Now if only I could describe this bird shape with functions...

Top Ten Fun Math Books

I wrote about my favorite math books for the Nerdy Book Club site. It's up on their site now.

Have I left out anyone's favorite?

A *Math* Petition?! Yep.

Did you know that the U.S. Federal Government has a website called We the People, where you can post petitions?

This may be the first math-related petition I've ever 'signed'. 

Implement a Policy for Declassifying Discoveries by NSA Mathematicians

The NSA is the largest employer of mathematicians in the United States. Currently, the discoveries of those mathematicians in their official areas of research, being deemed potentially critical to national security, are indiscriminately classified for an indefinite period, with limited circumstances for declassification.
It is requested the White House press the NSA for an expiration policy for the classification status of non-applied discoveries and instituting an expiration for gag order patents in the interest of furthering American academia and industry advancement and in the interest of crediting the discoveries of our nation's talented NSA employees.

Sign here.

Calculus: Anti-derivatives and Area Under a Curve

The textbook we use (Briggs), and I think most of the textbooks I've used in the past (including Stewart and Thomas), introduce anti-derivatives before area under a curve. So they show students the notation for indefinite integrals before showing them the notation for definite integrals. I think this is a BIG mistake.

Here's what happens...

∫ f(x) dx (aka indefinite integral) means find all functions F(x) so that F'(x)=f(x). 

(Why does it use that funny symbol? Why does it have that dx part at the end? Hard to explain without referencing a connection that hasn't been made yet, isn't it?)

And then we start thinking about areas under curves and use a notation that's almost the same.

∫₁² f(x) dx (aka definite integral) means find the area which is between f(x) and the x-axis (area below the axis counts as negative), and between x=1 and x=2.

Seeing almost identical notation and names, we're going to assume that these two act the same in some ways. Students are going to expect anti-derivatives, even though it's area we're talking about here. So it's not much surprise when the Fundamental Theorem of Calculus tells us that to find area we can use anti-derivatives.

Wait! That should be a surprise. It's kind of amazing, isn't it? Derivatives give us slope. Why would going backwards in that process give us area?! Seems to me that's a big one we need to meditate on for a while.

This semester I knew I wanted to connect the new ideas with the position, velocity, and acceleration problems, so I introduced anti-derivatives first. And, I showed the indefinite integral symbol. Oops! I shouldn't have. If I had held off, I believe the meaning of the definite integral would have taken hold better in my student's minds.

Until this semester, I've followed the textbook pretty closely, so my way around this problem has been to introduce the 'Area Function' without using this notation. I found this idea/project in a book put out by the MAA.  I've revised it a lot over the years, but the original author, Charles Jones (of Grinnell College) still deserves credit for getting me started in this direction. (I wish I could figure out how to thank him personally, but he doesn't seem to be at Grinnell College these days, and google gives me lots of people with that name.)

I've put a pdf of the project here. If you'd like my Word file, just email me (mathanthologyeditor on gmail).

We've started that project, and it's going well enough, but I realized that if I hadn't introduced the indefinite integral, we'd be better off. Next semester I'll get that right.

Tomorrow we wrap up the project, and I clarify the implications of the Fundamental Theorem. Cool stuff!

Monk Climbing Mountain Puzzle

Have you seen this puzzle?

A monk climbs from the base of a mountain to its top on the one narrow path up and down, sleeps in a hut at the top, and then descends again to her monastery the next day. She leaves at about 6am on both days, and arrives around 6pm on both days. She stops for a break whenever she feels like it.

Will there be a time of day where she’s at the same spot on both days?

Factor Diagrams

A while back I reviewed You Can Count on Monsters, a delightful book showing monsters built from the prime factorizations of each number 2 through 100. (1 is in the book, but is sad, since it can't be made from primes.)

There are now lots of other takes on this idea. Brent, at The Math Less Traveled, made some gorgeous factor diagrams back in early October. When he posted them, many of his readers took them as inspiration to do more. One made a factor tango. Brent was then inspired to  improve on his own diagrams.  Here's a partial picture of what's he done:

He says he'll be making posters and t-shirts. I think this would make a great poster for a math classroom. Maybe I'll get copies for some of my colleagues.