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In Art

Fibonacci: When art imitates nature

On Monday, I introduced you to Elizabeth Perkins, an up-and-coming glass artist in Seattle. (She also happens to be one of my former students, but that is mere coincidence. I take no credit whatsoever for her success and talent.) In her interview, she mentioned that she depends on the Fibonacci sequence to develop some of her annealing programs, or processes for cooling the glass so that is remains structurally sound.

But what the heck is a Fibonacci sequence?

Well, it’s a pretty cool list of numbers. And it’s also really, really easy to figure out. See for yourself:

0, 1, 1, 2, 3, 5, 8, 13, 21, ?

What’s the next number?

I’ll give you a chance to think about it.

Need a hint? Pick any number in the list (except for the first 0 and first 1), and look at the two numbers before it.

Get it yet? (The correct answer is 34.)

The Fibonacci sequence is generated by adding the last two numbers together to get the next number. Take a look:

0 + 1 = 1

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

8 + 13 = 21

13 + 21 = 34

Now that you know this rule, you could conceivably add numbers to this sequence until you got bored or exhausted (which ever comes first).

The fellow who discovered this sequence was, you guessed it, Fibonacci — an Italian mathematician and philosopher who was reportedly born in 1175 AD. But to be honest, his sequence is not the greatest contribution Fibonacci (or Leonardo de Pisa) gave to humankind. In fact, he is the father of our decimal system. Yep, the fact that you can count the $5.23 you have in your wallet is due to a guy whose real name we don’t even know for sure.

But I digress.

The Fibonacci sequence isn’t just an easy and cool math fact. It’s cool — and really, really important — because it shows up everywhere. Here are just a few examples:

If you count the petals of various species of daisies, you’ll get one of the Fibonacci numbers.

The length of the bones in your wrist and hand are a Fibonacci sequence.

The spiral of a pineapple is arranged in Fibonacci numbers.

Branches of a tree grow in a Fibonacci sequence (one branch, two branches, three branches, five branches, and so on, moving up the height of the tree).

The gender of bees in reproduction mirrors the Fibonacci sequence.

And then there’s art. Art loves the Fibonacci sequence. Since the Greeks formalized what is beautiful in architecture and paintings, this little list of numbers has been front and center in a variety of artistic fields.

For example, this seven plate print is gorgeous and also represents something called the golden spiral. The sides of each square (starting in the center with the smallest squares) correlate to the numbers in the Fibonacci sequence. So, the smallest square has side length of 1 unit, the next largest is 2 units, the next is 3 units, the next is 5 units, etc.

Cool huh?

It gets better. Remember the lady with the mysterious smile? Leonardo da Vinci was fascinated by mathematics, and some folks have noticed that his lovely lady’s facial characteristics follow the path of the Fibonacci sequence.

Image courtesy of www.shoshone.k12.id.us

Do you see how the squares line up with the base of her eyes and bottom of her chin, and surround her nose perfectly?

So there you have it. What we see as beautiful could very well be because of mathematical wonders like Fibonacci’s sequence. And as Beth the glass blower shows, this magical little list of numbers is useful in the science of making art as well.

Earlier this year, I posted a really, really cool video about the Fibonacci sequence in nature. Check it out here.Save

In Personal Finance

Trim Your Spending with Percents

The first step to becoming more financially stable is writing down what you spend — and being honest about it. But what happens when you subtract your expenses from your income, and you’re in the red? Pouring yourself a stiff drink may be a first step, but it’s not going to solve the problem for you. Instead, you’re going to have to put on your big-boy or -girl pants and get down to the business of trimming your spending.

But one of the tough parts about budgeting is making reasonable assumptions about what you should be spending on any one category of your budget. Does it make sense to spend 50% of your income on housing? Should you cut your monthly savings?

Our brains are funny little organs. We can convince ourselves that we must have that huge flat-screen television set or we deserve to go out for drinks with the girls every Friday night. But the numbers don’t lie.

Math can help keep you honest about what you’re earning, spending and putting away for a rainy day, retirement or when you decide that you’d rather be a writer than an advertising sales executive.

Each family or person is different, of course, but there are some great guidelines that can help you see if you’re on track. Here are some examples:

Housing should cost no more than 28% to 33% of your monthly gross income.
Groceries should account for about 18% of your monthly gross income.
You should be saving between 10% and 20% of your monthly gross income.

This is one of those situations when math can really help you lower the emotional impact of your decisions. Knowing what is reasonable to spend on these items can make it easier for you to actually make the changes.

So let’s say you’ve tallied your income and expenses and come up short. (No wonder your credit card bills are so high!) You gross $3,127 each month, and your rent is $750 each month. You spend about $650 on groceries and meals out each month, and you try to put away about $100 into savings.

Of these expenses, what should you cut? Let’s take a look. The experts estimate that your housing should cost no more than 28% to 33% of your monthly gross income:

28% of $3,127

0.28 x $3,127

$875.56

33% of $3,217

0.33 x $3,127

$1,031.91

Given your monthly income and the experts’ guidance, you should be spending between $875.56 and $1,031.91 each month on housing. Your rent is much lower that that, so unless you’re having your living room redecorated by Martha Stewart herself, you should be good to go in that category.

On to groceries:

18% of $3,127

0.18 x $3,127

$562.86

But you’re spending $650 on groceries and eating out each month. Clearly this is where you can cut some of your spending.

Finally, take a look at savings. While you could zero this out, so that you can pay off some debt, it’s probably not a good idea to forgo savings altogether. Besides, didn’t all of our parents preach about having a nest egg? (In fact, financial experts recommend that we have the equivalent of at least 4 months of our salary tucked away — just in case.) Building your savings takes discipline and time. And there’s no better time than the present to get started.

But how are you doing now, according to the expert guidance?

10% of $3,127

0.10 x $3,127

$312.70

20% of $3,127

0.20 x $3,127

$625.40

Hold the phone. With your measly $100, you’re not even close to what is recommended. Perhaps you could cut back on your clothing budget, so that you can actually retire on time or have a safety net if your job suddenly goes poof!

I’m the first to admit that these suggested percents are not the be-all-end-all of budgeting advice. Each one of us has extenuating circumstances to consider. But why not start with the math? In terms of what we’re spending, saving and earning, the numbers don’t lie.

P.S. For the really diligent among us, there’s something called the 50/30/20 budget: Must-have expenses (housing, food, insurance, etc.) should account for 50% of your income after taxes, while 30% should be “wants” and 20% should be savings. The trick here is deciding what is actually a “need” and what is really a “want.”

Using these percents, how are you doing with your monthly spending? Calculate what you should be budgeting for housing, food and savings, and then compare those results with your actual spending and savings. Tell us how you stack up in the comments section — and best of all, whether the result is surprising.

In Personal Finance

Budget basics

your New Years resolution is to save money — or spend less — most financial folks will tell you one thing: you’ve gotta have a budget. This means figuring out what you earn and how to spend those earnings. Budgets can be complex or simple. It all depends on what you are comfortable with. (Personally, I go for simple, because all of those details keep me from maintaining good finances. But if I needed to pay off a lot of debt or save a good amount of money, I might suck it up and look at every single penny.)

For today’s post, I thought I’d just print an excerpt of my book, Math for Grownups. Chapter 8 is called “At the Bank,” and it deals with money issues (aside from shopping, transportation or housing, which are covered in chapters 1, 2 and 3).

It’s New Year;s Day, and Darrel is pondering his resolutions over a bowl of black-eyed peas. For sure, he wants to reach level 65 in Purple Heart: World at War. And he wants to ask out that cute girl in the apartment next door.

But Darrel is also sick and tired of worrying about money. He’s got a good job as a computer programmer, but for some reason, he’s still ending up with too many bills at the end of the month. Last year, he had to sell is first-edition Spiderman comic to pick up a little extra cash. He knows he needs to add a really, really boring New Year’s resolution to his list: keeping a personal finances budget.

He vaguely remembers what his high school consumer math teacher told him about budgets. At least he remembers there are three parts: income, regular expenses, and occasional expenses. His income should be greater than all of his expenses put together.

He writes the name of the month at the top of a piece of paper, January, and adds his current monthly income: $2,655.

He’s careful to put his take-home income, not his before-tax income, because that’s all he can spend.

Now he brainstorms all of his regular expenses, including his weekly comic store purchases. Some of his expenses, such as his electric bill vary a bit form month to month, but he adds up the last year’s worth and divides by 12 to get a monthly average.

Expenses
Item	Cost	Item	Cost
Rent	$800	College loans	$200
Electricity	$145	Gas	$100
Water	$21	Comics	$100
Cell	$80	Groceries	$400
Internet	$42	Entertainment	$200
Satellite	$100	Clothing	$100
Car payment	$360
Total		$2,648

So far, so good. It looks like Darrel is living within his means, but what will happen when he adds in his occasional expenses? He brainstorms again, consulting his online banking records for guidance.

Occasional Expenses
Item	Cost	Total per year
Car insurances	$450 every quarter	$1,800
Comic book conventions	$4,200 per year	$4,200
Professional association dues	$500 per year	$500
Dojo fees	$275 per semester	$550
Gifts	$170 per year	$170
Total	$7220

He divides that total by 12 to get his average monthly expense: $601.67.

Darrel adds his regular and occasional expenses together: $2,648 + 601.67 = $3,249.67. That’s more than his monthly take-home pay! He’s going to have to cut back. It takes Darrel only a few moments to recalibrate his budget. He’s going to reduce the number of comic book conventions he goes to and cut down his satellite television expenses. With that, he notices that he can put some money each month into his languishing savings account. And if, at the end of the year, he gets that raise he’s been expecting, he can put even more away for a rainy day.

This little bit of math gives Darrel a boost of confidence — enough confidence that he picks up the phone and calls his cute neighbor.

Do you use a budget? If so, what kind? And how has it helped you manage your finances?

In Home

Ready, Set, Organize!

When I decided to organize my junk drawer two weeks ago, I did what most folks do — I purchased a drawer divider set with a variety of different sizes. The idea is to group like things together. The pencils go in one section, pens in another. Littler compartments hold paper clips and Box Tops. And the biggest container is for my precious scissors, which seem to go missing at least once every other day.

In fact, this is the No. 1 tennet of organization: A place for everything and everything in its place. If I have a designated spot for my daughter’s erasers, they won’t be strewn around my kitchen counters or tossed into the silverware drawer. (And she won’t be screaming in a fit of last-minute homework, “I can’t find an eraser!)

At least that’s the idea.

And that idea is as old as dirt. In fact, it has its roots in mathematics, specifically set theory, which wasn’t formalized until Georg Cantor, a German mathematician, published an article on the subject in 1874. This blew the socks off of the mathematics community — mainly because he proposed that there are two kinds of infinities.

But I digress.

Kindergarteners learn about set theory, when they circle like things on a worksheet. And many parents probably wonder why this is such a big deal.

In short, set theory is the basis of our numerical systems — among many other things. Mathematics craves order. Knowing why things are alike or different can help us solve problems quickly and effortlessly. Just like knowing where my scissors go (and putting them there) makes it easier for me to find them later on.

As an example, let’s look at the set of whole numbers.

{0, 1, 2, 3, 4, 5, 6, 7, … }

(Okay, just so no fancy-schmancy mathematician jumps down my throat, I have to note here that there is some disagreement about whether 0 belongs in this set. But for most of the rest of the world, that’s a point not worth arguing about.)

When you know the set of whole numbers, you can determine whether or not a number is in that set. For example:

0.25 is not a whole number

60% is not a whole number

π is not a whole number

-17 is not a whole number

But: 6,792,937 is a whole number

But why do you care? Honestly, I think the biggest reason is so that you can talk about math. In this case, set theory tells us the difference between whole numbers, integers, decimals, rational numbers, etc. — even if you don’t remember what all of these are.

(And those of us who know a little bit about math also know that whole numbers are in the set of integers, which are in the set of rational numbers, which are in the set of decimals.)

So this is how math is like organizing. Both depend on set theory.

I’m not saying that you have to be organized to do math. Lord knows I’m not. But the underlying organization of math points to big clues about how it’s done. Even more basic sets, like geometric shapes can apply in our everyday lives.

The bottom line is this: If you think you can get your house or office or car organized (and I believe you can!), you can certainly organize all of what you know about math and put it to good use. That way, you’ll always know where your area of a triangle is.

How do you think about the structure of numbers or shapes or arithmetic operations? This points to your intuitive understanding of set theory. Share your thoughts in the comments section!

In Home

Getting organized bit by bit

“My house is a disaster.”

How many times have you uttered these words or heard someone else say them? You and they are not alone. Getting organized is one of the most common New Year’s resolutions. But like losing weight, it’s easier said than done.

But how do you manage this daunting task? If you’re inclined to take a week off of work, with high hopes of a sparkling, organized home after five long days, you may want to reconsider. If you’re not already organized, why would you want to spend so much time cleaning out your linen closet and kitchen cabinets?

On this point, the experts agree: a little goes a long way. So most suggest that devoting only 15 minutes a day to organization can yield big benefits. Let’s take a look at the numbers.

If you devote five days, for (let’s be generous) 10 hours a day, you’ll end up working 50 hours total, right? (That’s 5 days x 10 hours or 50 hours.) And you’d probably also have a sore back and a week’s worth of vacation lost to your label maker and plastic bins and lids.

But what if you committed to 15 minutes a day, 5 days a week? How much time will you have spent?

15 minutes x 5 days = 75 minutes

75 minutes ÷ 60 minutes = 1.25 hours (or 1 hour and 15 minutes)

Gosh, I spend more time in a week figuring out what’s for dinner.

So what if you started on January 1 and stuck with it throughout the month?

There are 22 weekdays in January

15 minutes x 22 days = 330 minutes

330 minutes ÷ 60 minutes = 5.5 hours

That’s less than the time it would take for you to watch the first two films in the Lord of the Ringstrilogy!

So let’s take this a bit farther. If you managed to keep this resolution for an entire year, how much time will you have spent organizing? Let’s assume there are 250 workdays in the year. (You’re not going to organize on a holiday are you?)

15 minutes x 250 days = 3,750 minutes

3,750 minutes ÷ 60 minutes = 62.5 hours

So by devoting a mere 15 minutes a day to organizing, you can end up spending more time over the year than if you took a week off and worked on the task for 10 hours a day. Plus, I guarantee you’ll be much more relaxed.

But what can you accomplish in 15 minutes? Here’s a short list:

Cleaning out your junk drawer
Going through seasonal clothes and deciding what to give away, toss or keep.
Alphabetizing your spice rack.
Culling through your kids’ artwork and filing or scanning special pieces.
Scanning your bookshelves for titles you’re ready to part with.
Setting up a spot for your mail, keys, purse and jacket.

By the end of one week, you could have a tidy junk drawer, trimmed summer wardrobe, room on your bookshelves and a regular spot for your keys. By the end of the year? Who knows what you could accomplish!

Have any organizing tips to share? Post your ideas in the comments section. I’ll bet I (or someone else) can find the math in that technique!

In Health

Watching the Weight Go Down

I suspect I’m not alone in one of my New Years resolutions: to eat better, exercise more and lose weight. (Ack! Did I just write that out loud?) Like other women in their mid-forties, I have found my metabolism screeching to a halt and my weight creeping up and up. So last night, I launched my most recent — and hopefully last — attempt at getting into better habits.

But I’m not fooling myself. This is a long process with a lot of little steps, some forwards and a few backwards. My issue is staying motivated. I do well for a while, and then I slip up — and eventually give up. So, I’ve got a good plan that should allow me to make incremental changes and leave lots of room for mistakes. And to accomplish this, I’ve got both measurable and soft goals.

Sure, I want to eat better (I can count each serving of veggies I eat or glass of water I drink.) And I want to exercise more. (I can count my hours at the gym or steps I take from my car to the grocery store.) But for me, those are big changes that will include a lot of frustrating missteps. In order to stay focused, the real measurement will be my weight.

My goal is to lose 25 pounds. And I’ll track this by weighing myself once a week.

So how long will it take me to lose the weight? This is where the math comes in. Here’s what I know:

I want to lose a total of 25 pounds.
I will probably lose between 0.5 and 2 pounds each week.

Does this mean I’ll be sitting on a beach in a string bikini in August? (That’s a joke. I’ve never worn a bikini in my life.) Let’s look at the math.

A half pound and two pounds is a pretty wide spread, so based on past experience and my inclination to be more conservative, I’m going to estimate that can lose about a pound each week. So I can reasonably expect to lose all of the weight in 25 weeks.

Whatever you expect to lose in a week, the math is simple:

total weight lost goal ÷ loss per week = number of weeks

25 pounds ÷ 1 pounds per week = 25 weeks

So if you think you can lose 2 pounds per week, it’ll take you 12.5 weeks to lose 25 pounds:

25 pounds ÷ 2 pounds per week = 12.5 weeks

But there’s one more step I need to take. I don’t think in terms of weeks. My brain focuses on months. How many months will it take me to lose the weight?

Again, I’m going to estimate. While there are approximately four weeks in each month, that’s not an exact figure (except in February during a non-leap year). But since I’m not measuring out medication or figuring out how much to send into the mortgage company or solving problems for my eighth grade math teacher, I don’t have to be exact. So I’m going to go with four weeks in a month.

I already figured out that I can probably reach my goal in 25 weeks. To find out how many months that is, I can just divide by 4 (the number of full weeks in a month):

25 weeks ÷ 4 weeks per month = 6.25 months

Ta da! I can reasonably expect to lose this weight in six months. That means if all goes well, I should be at my goal by June.

Like me, are you hoping to lose weight in 2012? Do the math to see when you’re likely to reach your goal. And if you want to share, feel free in the comments section. (It’s scary, but you can do it!)Save

In Holidays

Talking Turkey

I’m taking it easy this week (ahhh!), and so I’ve brought you an excerpt from my book Math for Grownups. (Check it out for more great ideas on using math in your everyday life.) Happy Thanksgiving!

As any experienced cook will tell you, timing is often the most difficult skill to master in the kitchen. Nobody wants to sit down to a meal of overdone fish, cold broccoli, and room-temperature biscuits. (The butter should melt into the flaky layers, you know?)

Figuring out how long a dish should bake, roast, or boil is the first step to presenting a carefully choreographed dinner. And for many novice or not-so-frequent home chefs, a giant turkey is the most daunting of all entrées.

Sure, you can count on the pop-up timer. These come with some turkeys, or you can buy one separately. But you’ll still need to know when to put the bird in the oven—and when to start boiling the potatoes.

And there’s also the thawing time. Buying a frozen turkey means allowing time for it to defrost, which is probably a lot longer than you think!

But you don’t need Julia Child or a semester at Le Cordon Bleu to figure any of this out. Thawing times and cooking times depend on the turkey’s weight.

It’s your first Thanksgiving with your new husband, Tom. And your mother-in-law will arrive just in time for the 6:00 P.M. dinner. She’s bringing pecan pie, stuffing, and homemade rolls. You’re in charge of all the rest—including the turkey. You’ve ordered a 12-pound bird, which you’ll need to thaw in the fridge before roasting. When should you pull it out of the deep freeze?

You know from your sister’s horror stories that you can’t cut corners by thawing the bird on the counter. Unless you want to host the Thanksgiving-dinner-when-everyone-got-Salmonella, your best bet is to defrost the turkey in the refrigerator. The United States Department of Agriculture(USDA) says to allow 5 hours of thawing time per pound. They oughta know, right?

You’ve bought a 12-pound turkey. How long should you allow for thawing?

It’s a simple problem, really. Just multiply the number of pounds by 5—the number of hours needed to thaw each pound.

12 x 5 = 60

So you need to put the turkey in the fridge for 60 hours in order to thaw it. But let’s think a moment. Does this mean 60 hours before dinner is served? Nope. The USDA also says that serving raw poultry is a big no-no, so you’ll also need to roast the bird.

If your oven is set to 325ºF, the USDA recommends roasting an unstuffed turkey for 2¾ to 3 hours. They’re the experts on avoiding food-borne illnesses, so you decide to follow their recommendations.

With a little time for resting—the turkey, not you—and carving, you estimate that it will take 3 to 3¼ hours to get the bird from the fridge to the table. You’ll need to add that to the thawing time in order to figure out when to pull the turkey out of the freezer.

60 + 3¼ = 63¼ hours

Clearly you’ll need more than a day, but how much more? There are 24 hours in a day. How many 24s are there in 63¼? You can use a calculator, but that could be confusing. Instead, try some mental math.

To make things easier, forget about the extra ¼ hour (or 15 minutes). You can add that on to the end. Working with whole numbers is much easier.

It looks like you’ll need at least 2 days. That’s because 24 times 2 is 48, which is less than the total time you have figured out. Will you need a third day? You can subtract to find out.

63 – 48 = 15

So 2 days and 15 hours (plus the extra 15 minutes) ought to do it. But that doesn’t tell you what time to start defrosting the turkey, does it?

Remember, your dinner starts at 6:00 P.M. Fifteen hours before that is 3:00 A.M., and another 15 minutes before that is 2:45 A.M. So you will have to take the turkey out of the freezer at 2:45 A.M. on the Tuesday before Thanksgiving.

Because you’re doing all the cooking, you decide to let Tom get up to move the turkey from the freezer to the fridge. You set his alarm on Monday night and settle in for the last good night’s sleep of the week.

Do you have any Thanksgiving cooking horrors to share? Do tell (in the comments section)!

In Math for Grownups

It’s a contest!

On Tuesday, I appeared on Midday with Dan Rodricks, an hour-long, call-in radio program on Baltimore’s WYPR. At the top of the show, Dan asked listeners to solve a real-world math problem. (Download the program here, to hear the entire show and what problem he offered.) I was so surprised at the number of people who called or emailed in with their answers — they loved it! So I thought I’d try it here on my blog.

Welcome to my first Math for Grownups contest! Here’s the background.

Maybe you’ve seen this image on Facebook or somewhere else on the web:

Get it? Funny, huh?

There are so many different ways that this fellow could have represented $536.49, and I think this is one of the misunderstood beauties of math. We were often taught that there is only one way to do a problem — but for the most part, there are many, many different ways to arrive at the correct answer.

And that’s the beauty of being grownups. We get to choose our own paths, right?

And here are the contest deets:

How would you have expressed $536.49? Get creative. Get complicated if you want. The only catch is what you describe has to equal approximately $536.49 (in other words, rounded to the nearest 10th or cent).

Here’s an example: (67 x 8 ) + (0.7)^2. (^2 means “squared,” which I use because it’s not easy to use superscripts in these blog posts.)

And here are the rules:

1. Post your response in the comments section here or on the Math for Grownups facebook page, by Monday, October 24 at midnight EDT.

2. Your response must be unique. That means, you must read through the other responses before posting yours. If there are two or more comments with the same correct response, I will accept only the first response.

3. You can respond up to five times.

4. The winner will be chosen randomly from all of the correct responses. (In other words, if your math doesn’t work out, your name will not be entered into the drawing.)

5. If you have five correct answers, your name will be entered five times.

6. One winner will receive a signed copy of Math for Grownups and a Starbucks gift card valued at $[(4 x 2) – (8-10)]. (Figure that out!)

7. I will contact the winner for his or her address so that I can send out the gift card.

Good luck!