Archive for the ‘Brain Workouts’ Category

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The Mathematics of Patterns

September 2, 2010

Every now and then I put in the DVD for a lecture series on discrete mathematics and watch one. I like discrete math because a) it’s not calculus and b) it’s all about patterns, so it’s like doing a puzzle. I can actually wrap my head around discrete math. It suits the regimented side of me.

Anyway, today’s lecture started off about recognizing patterns in series of numbers added together. I knew (half my life ago) that there was a shortcut, but I never knew what it was or even thought about it after high school.

Professor Arthur Benjamin* showed an example series. Say you want to add up consecutive odd numbers starting with one. What’s a quick way to determine the sum without physically adding them all? You try to find the pattern.

1=1

1+3=4

1+3+5=9

1+3+5+7=16

1+3+5+7+9=25

and so forth.

If you were to assign the number n to mean the number of numbers (sorry if that’s confusing) in the series, you’d start to see a pattern:

if n=1, then the series is 1=1, which is the same as 1=1²

if n=2, the series is 1+3=4, which is the same as 1+3=2²

if n=3, the series is 1+3+5=9, which is the same as 1+3+5=3²

and so on. So we figure out that if we add together n consecutive odd numbers starting with one, the sum will be n².

Let’s say we have the series:

1+3+5+7+9+11+13+15+17+19+21

We can count all the numbers, and we see there are 11. Eleven squared is 121, so the sum of that series is 121.

But what if you have a series that’s super long, and you don’t want to count how many numbers are in it? You find the pattern for the last number in the series and how it relates to n.

Again, the series:

for 1 , n = 1

for 1+3, n = 2

for 1+3+5, n = 3

for 1+3+5+7, n = 4

so,

when n = 1, the last number in the series is 1

when n = 2, the last number is 3

when n = 3, the last number is 5

and when n = 4, the last number is 7

If we puzzle it out, we’ll notice that by doubling n and subtracting 1, we’ll get that last number. So, for example, when n = 4, we get the last number, 7, by doubling 4 to make 8 and subtracting 1.  The last number is 2n-1.

So now we can write the series this way:

1+3+5+7+…+(2n-1)=n²

I now have the shortcut. If I want to know what the sum of all odd numbers up to 99 are, I don’t have to write them out. I can say, “I know the last number is 99, so 2n-1 is 99. I can add one to make 100, divide it by 2 to get 50, and now I know n is 50. So the sum is 50 squared, which is 2500.”

I determined to figure out on my own the pattern for summing consecutive even numbers starting with 2, and in about 15 minutes of drawing it out, I did it! The answer is:

2+4+6+8+…+2n=n(n+1)

If I want to know the sum of all even numbers up to 100, I know that n=50, which makes the sum 50 times 51, or 2550.

I’ve never figured out a mathematical pattern by myself before, so I just wanted to share. It was kind of exciting!

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*Arthur Benjamin is a wonderfully enthusiastic teacher. You can check out his “Mathemagician” web page here.

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Just in Case You Were Thinking About It

August 25, 2010

I love this sign. It’s outside the full-scale replica of the Parthenon in Nashville, Tennessee. What would Socrates say?

P.S. Am looking through vacation photos now that the kids are in school. I’ll eventually write about The Bath.

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Ego Sum īgnāva

July 26, 2010

I haven’t written part two of the essay. Part of the reason is that I’ve been distracted by something more important, but the other part is that it’s hard and I’m still struggling to overcome my academic laziness.

I am not proud of this. I like to hope that announcing my weakness will motivate me to overcome it. I see it as similar to telling everyone I’ve started a diet. Which I haven’t. Just to be clear. So no weight comments, please.

I was scrubbing the grout on the kitchen floor yesterday and entertaining myself by listening to a lecture about the Middle Ages. (Synopsis: Black Death=Bad; Printing Press=Good)

I was sort of tuning in and out when the professor mentioned the shift in scholarship from the Continent to Britain. For a long time all of the great church scholars had been from Italy, but all of a sudden you had guys like Thomas Aquinas and William Ockham coming out of England and Ireland and taking over the academic world. Why was this?

It’s because they didn’t speak Latin. Rather, they didn’t understand Latin natively. Italian is derived from Latin. French is derived from Latin. Spanish is derived from Latin. But English is not. English is a Germanic language that is completely unrelated to Latin. The theory is that Continental scholars didn’t have to work as hard in their studies (all in Latin) because they already spoke a language that was very close to Latin.

They were lazy students, and the Brits took over the world because they had to study harder to learn anything.

I contemplated this point as my aunt and I met with her surgeon this afternoon. He moved to the U.S. from Vietnam when he was in his teens, learned English, graduated from a local high school, and got a bachelor’s degree in Electrical Engineering. He spent five years at IBM before going to medical school and becoming a surgeon. He’s been in practice 10 years.

Obviously this man is brilliant. More amazingly (to me, anyway) he’s an incredible student. I can’t imagine ever being motivated enough to get the education required for either an engineering or a medical career. He’s done both.

I’ve read that graduate students in science and engineering at the top American universities mostly come from foreign countries where English is not the native language. I’ve observed that the three local M.D.s I’ve talked to this week grew up speaking Vietnamese, Turkish, and Spanish.

It has me wondering: is there some truth to this theory that people who are forced to overcome a language barrier make better students of subjects that require intense study? Are we undergoing a cultural shift whereby most of America’s doctors and engineers will speak English as a second language?

Something for me to ponder as I avoid serious thinking.

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A Two-Part Essay

July 21, 2010

In 1942 in California, John Steinbeck wrote a novel called The Moon is Down. Meanwhile, World War II raged in Europe, North Africa, and the Pacific.

1942 was a dark year. The Reich was rising. Storm troopers under the command of Adolf Hitler’s National Socialist Party had captured and now controlled Austria, Czechoslovakia, Poland, Holland, Belgium, Norway, France, Hungary, Romania, Yugoslavia, and Greece. A little Jewish girl in the Netherlands called Anne Frank started a diary. That summer she would go into hiding with her family while Jews all over Europe would be deported to concentration camps. At Auschwitz, Nazis began gassing the prisoners.

The first American troops arrived in Europe in January. German subs attacked the coastline of North America from Canada down to Mexico. Japan was busy taking over the Pacific. In April they would conquer the Philippines and instigate the Bataan Death March. Japanese subs attacked Australia and islands all over the Pacific.

People were desperate. The outcome of the war was unclear.

John Steinbeck was 40 years old. By 1943 he would be a war correspondent for the New York Herald Tribune, but in ‘42 he contributed to the war effort by writing a propaganda novel, one intended to give hope to the people of occupied Europe.

In The Moon is Down, an unnamed little European town is captured in a nearly bloodless coup by an invading force, and now the people must endure occupation. Resistance is futile, the new leaders tell them. We can all live pleasantly together if you just follow our rules. But of course the people are not happy to have their guns seized, to have their homes occupied, to be forced to work in a coal mine for the new commanders, and to face imprisonment or death for failing to follow orders. They think of themselves as free men, and they resent their captors.

What makes this novel interesting and different from most propaganda is that the occupying force are portrayed as real people, not all of whom buy into what they’re doing. In particular, Captain Lanser of the invading force has his doubts. He has lived through war before and he knows that there can be no bloodless occupation of a country, no suppression of a free people without revolt. He can see the difficulty that will come, and he is tired, and he is frustrated, and he tries to be fair and kind to the townspeople, but ultimately he sees himself as a cog in a machine that is orchestrated by The Leader. He must do as The Leader wills. His duty comes first, regardless of his personal moral standards. He has never been free.

And so he watches without surprise or horror as the townspeople implement their resistance, picking off armed troops one at a time with stones or fists, sabotaging rails and bridges, and performing their required work in the coal mine slowly and badly.

His troops become discouraged. They never feel safe. They receive no human warmth from the townspeople. They know they are hated and they want to go home.

And all the time the Captain must press on because The Leader wills it. He knows his effort is fruitless, that it is a waste of lives on both sides, and yet he cannot leave. He has no power. He is required to occupy the town.

In the final scene, the town Mayor and Doctor are arrested and held hostage, to be executed if another act of sabotage is committed. They know the sabotage will happen and that they will be executed, and while they’re not happy about it, they are resigned to it. They also know that without their leadership the resistance will continue under new leaders, and when those are executed, new leaders, and so forth until the end. It is a fundamental difference between them and their occupiers, who are dependent on a handful of men in leadership position: if those leaders were gone, the occupiers would not know what to do. But free men will rise and lead themselves.

In an interesting little speech, the Mayor remembers back 46 years to his time in school with the Doctor and how they had to memorize Plato’s Apology, the defense that Socrates gave to the senate at Athens when he was condemned to death for treason. The Mayor stumbles over the words as he recalls the speech, and Captain Lanser of the occupying force corrects him, for he, too, memorized Plato in his youth. But the former feels the speech as a free man and the latter only hears it as pretty, meaningless words.

As Captain Lanser tries to convince the Mayor to rein in his people — as though the Mayor had the power to do so — explosions are heard outside. The sabotage has begun again. The Mayor voluntarily leaves with a soldier to face his execution. He pauses to quote Socrates’ final words to his old friend, the words Socrates spoke right before he drank the hemlock.

In the doorway he turned back to Doctor Winter. “Crito, I owe a cock to Asclepius,” he said tenderly. “Will you remember to pay the debt?”

Winter closed his eyes for a moment before he answered, “The debt shall be paid.”

[The Mayor] chuckled then. “I remembered that one. I didn’t forget that one.” He put his hand on Prackle’s arm, and the lieutenant flinched away from him.

And Winter nodded slowly. “Yes, you remembered. The debt shall be paid.”

The reaction to Steinbeck’s novel was immediate and powerful. The Nazis banned it, and Mussolini proscribed death to any Italians who possessed it. In occupied France and Norway, it became a rallying point for the underground resistance, who translated and distributed it all over Europe. It became the best-known work of American literature in Soviet Russia during the war.  In 1943 it was made into a movie. After the war Europeans commented that they couldn’t believe Steinbeck was able to write so realistically about their experience. It was as though he had been there.

I have recently and coincidentally read this and another novel published in 1942, and I’m struck by how the war influenced both writers and led them to opposite conclusions about the nature of man.

In 1962 Steinbeck was awarded the Nobel Prize for Literature. In his speech he said, “I hold that a writer who does not believe in the perfectibility of man has no dedication nor any membership in literature.”

Next up, a novel by a writer who did not believe in the perfectibility of man: The Once and Future King by Mr. T. H. White.

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Babylonian Math

June 17, 2010

Until I can locate my camera’s power cord, there will be no vacation photos. Try to restrain your disappointment. ;)

While the kids were at Vacation Bible School yesterday (a concept I find simultaneously delightful and mystifying given my secular upbringing — but that’s a whole other post) I spent an hour avoiding cleaning the house and instead watching the first two lectures in a series devoted to the history of mathematics.

I’ve really come around to liking math lately. It’s a foreign language I never bothered to learn well when I was in school. I’m slowly fixing that.

I especially like learning about ancient math and science. It’s riveting to discover that people 4,000 years ago were far more advanced and modern in thinking than I’d realized. The ancient Babylonians predicted eclipses and planetary alignments centuries into the future, they calculated the square root of 2 to six decimal places, they figured out a geometrical solution to quadratic equations, and they used the Pythagorean theorem 1,500 years before Pythagoras was born.

I wish I had learned in school that algebra is shorthand for something that can be solved using geometry — visually. And had been shown those visuals. I think it would have been a lot easier to grasp.

Take the Pythagorean theorem. The algebraic equation is

a^2 + b^2 = c^2\!\,

where a and b represent two sides of a right triangle and c is the hypotenuse. When we visualize it, this is probably what we see:

It doesn’t really tell us much.

But what if we think about it like this: a square with sides of length A will have an area that, when added to the area of a square with the sides of length B, will equal the area of a square with the sides of length C.

We’ll use the classic 3/4/5 triangle:

This makes more sense to me. And it’s how the Babylonians derived the theorem themselves. When the numbers of the three sides are all whole numbers we call them Pythagorean triples. The Babylonians calculated this triple: 4601 (squared) + 4800 (squared) = 6649 (squared). Crazy.

How about quadratic equations? Let’s say we’re staring at this quadratic equation in 10th-grade algebra and trying to figure out how to solve it:

x² + 2x = 15

Before this concept could be represented algebraically, the Babylonians figured it out geometrically. Let’s say you wanted to construct a rectangular building and you knew you only had 15 square units of ground space to work with. You know that you’d like to divide it into two rooms, and one of those rooms needs to have a wall that is 2 units long. The other room needs to be a square. How long will the walls of the square room be? You can draw it like this:

Can you see it?  x² + 2x = 15

Now you can split the rectangle with a known side of 2 units into equal pieces:

Next you move these two pieces around like this:

Now comes the part known as “completing the square”. You create a little square to fill in the missing space in the bigger square. We can tell from the drawing that the new square will be 1 unit by 1 unit.

You’re going to need to add an equal square to the other side to keep the equation equal:

The little square has an area of 1 unit. We can think of the big square this way: its area is the same as 15 plus 1. We can write it out showing that one side multiplied by the other side equals 15 plus 1:

(x + 1)(x + 1) = 15 + 1

We’ll write it out a new way:

(x + 1)² = 16

And now we need to take the square root of each side, something the Babylonians were good at doing.

(Square root, by the way, is the length of a side of the square that can be divided into equal squares of a given number. So 16 little equal squares can be combined to form a larger square with sides the length of 4. The square root of 16 is therefore 4. This becomes more complicated to do geometrically when you’re not using whole numbers, like say, finding the square root of 15. But it works for square roots that are whole numbers.)

Anyway, taking the square root of each side gives us:

x + 1 = 4

Now it’s simple to solve for the unknown. Simply subtract 1 from each side and:

x = 3

Now we know our square room is 3 by 3 units, and the total rectangle building is 3 by 5 units.

Algebraically speaking, x also could equal -5, but I don’t know if the Babylonians could calculate that. While I’m aware that there must be some use for -5 as an answer to this equation, it isn’t practical for the question I posed above about building dimensions, so it doesn’t much bother me whether the Babylonians could calculate it or not.

When I learned how to solve equations like this in high school, we did what is called “factoring”, meaning that you have to figure out what numbers can be multiplied together to get 15, and then insert them, one at a time, into the equation. One would be positive and the other negative, and so you would reason out 3 and -5 after trying them in the equation. It seemed so arbitrary to me at the time, and I had completely forgotten how to do it at all until I looked it up on the internet.

But I think I can remember how to complete the square. Pretty neat, huh?

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Byeya’s Irises

April 10, 2010

Somewhere close to 60 years ago my grandmother planted these “champagne” irises. They’ve survived droughts, freezes, and then 18 years of neglect after she died. More recently they survived being dug up and moved to my house.

I’ve been watching them, anticipating the day they would burst forth.

Today was a perfect spring day. I brought Augustine’s Confessions outside to read after it got too sunny to keep weeding. A small hill in my backyard proved magnetic after awhile, and I was drawn out of my shaded spot to lie there on my stomach and read.

Back in college, half my life ago, I used to lie outside on a hill reading in a sunny area between dorms, my friends to either side of me, listening to the music of Pearl Jam’s Ten blaring from a stereo propped in the open window of the boys’ dorm to our right combined with the music of romping 19-year-olds hitting volleyballs, throwing frisbees, and reveling in the freedom of new adulthood.

Today I listened to the music of neighborhood lawnmowers, the TV in the garage where my husband worked, and the shouts of my children romping with their neighbors in the yard next door, reveling in the freedom of a sunny Saturday with no homework.

“Time does not stand still, nor are the rolling seasons useless to us, for they work wonders in our minds,” Augustine said.

I took in the beauty around me, noting the prickle of last year’s grass thatch against my wrists, the contrast of the gentle warmth of the April sun soaking through my clothes and a light, cool breeze on my face, the visual punch of reborn plants bursting with foliage in the slanting light, and the smell of mulch and new growth.

“Yet were these beautiful things not from you [God], none of them would be at all. They arise and sink; in their rising they begin to exist and grow toward their perfection, but once perfect they grow old and perish; or, if not all reach old age, yet certainly all perish. So then, even as they arise and stretch out toward existence, the more quickly they grow and strive to be, the more swiftly they are hastening toward extinction. This is the law of their nature. You have endowed them so richly because they belong to a society of things that do not all exist at once, but in their passing away and succession together form a whole, of which the several creatures are parts. So it is with our speaking as it proceeds by audible signs: it will not be a whole utterance unless one word dies away after making its syllables heard, and gives place to another.”

And I saw what he was saying. We are part of a greater whole. We can’t see it because we are just parts, but for the whole to exist we must grow and eventually die, as did those before us and those yet to come. It is like we are the individual words of a spoken sentence, one that will not have meaning until all the words are spoken. It is bittersweet to see this beauty and to know its temporariness and yet it must be. We grow toward perfection in our mortal selves, and if we’re lucky we grow old before we perish. But always we are moving on, moving toward the true perfection of the whole.

Before I went inside I took a picture of Byeya’s irises. Today another part reached perfection.

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On Friendship

April 5, 2010

Friendship can only exist between good men.

This is what Cicero said in his letter called On Friendship. I decided to read it tonight after seeing it referenced in St. Augustine’s Confessions, which I’m working my way through this month.

Is it true that friendship can only exist between good men?

Well, to parse it some, I’d argue that friendship can exist between good women as well as between a good man and a good woman. And before you go and say that it’s 21st-century feminist mumbo jumbo that’s making me claim that Cicero meant only a man-to-man relationship I will tell you that he meant exactly that. He said so explicitly. Repeatedly. Only. Between. Men.

Moving on, do both parties have to be good? I got a kick out of his definition of “good”. Cicero was Roman, the speakers in his story are Romans, and while they acknowledge some indebtedness to the Greeks, they’re not going to blindly accept Greek philosophy. So where Socrates would have spent 20,000 words building an imaginary city to define “good”, Cicero (or his narrator, Laelius) snorts at the philosophers:

I do not, however, press this [definition] too closely, like the philosophers who push their definitions to a superfluous accuracy. They have truth on their side, perhaps, but it is of no practical advantage. Those, I mean, who say that no one but the “wise” is “good.” Granted, by all means. But the “wisdom” they mean is one to which no mortal ever yet attained. We must concern ourselves with the facts of everyday life as we find it — not imaginary and ideal perfections.

The argument Cicero makes as his letter drags goes on is that people who aren’t good can’t be good friends and therefore can’t have good friends. Or something like that.

Aside from a few snarky remarks, this was a pretty dry read. It ended up, in fact, being an occasion where the book itself became more interesting than the contents.

Take this excerpt from the introduction:

The evils which were undermining the Republic bear so many striking resemblances to those which threaten the civic and national life of America today that the interest of the period is by no means merely historical.

The edition I’m reading was published in 1909.

More fun are the notes a previous owner made in the margins, helpfully dated July 28, 1951. This passage was underlined:

There are people who give the palm to riches or to good health, or to power and office, many even to sensual pleasures.

And next to it, double-underlined and with an exclamation point is:

Tom!

So of course my imagination has drifted away from what Cicero had to say about friendship and toward whatever Tom must have done 60-some-odd years ago to merit double-underlining and immortality in my book. Was he into money or politics or *gasp* sensual pleasures? I want to know, RBJ (I only have your initials) … it must be a good story.

And now you see why I have trouble finishing tough reads.

But I did finish this one.

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Happy Easter

April 4, 2010

Did you know that St. Augustine was responsible for infant baptism? In his time (the 4th century AD) baptism was reserved until the last possible instant. You had only one chance to be completely cleansed of sin, so Christians in the 300′s held out until they were close to death, thinking that they had a better likelihood of achieving heaven if they were close to sinless at the time they met their maker.

Augustine reasoned that his delayed baptism led him to act worse than he otherwise would have. He lied, he stole, he took a mistress and fathered a child with her, and all of this was excused by his elders because he had not yet been baptized. He had no incentive to behave. Or so he said.

We went to the Easter Vigil service on Saturday night. It’s the biggest celebration of the year in the Catholic Church. (Christmas is secondary because without the belief in the resurrection, there’s no point in celebrating some prophet’s December birthday.)

It was our first time to bring the kids. Easter Vigil is a tough service because it’s very long — nearly three hours. But it’s a special service. It’s the service of adult baptism.

Seven years ago at Easter I converted to the Catholic Church. I was baptized a Methodist in my infancy, and when I attended services occasionally someone would be inspired to join the church. When that happened, the Reverend would welcome him to the front, ask him a few questions, and *bam* we had a new Methodist.

Not so with the Catholics. To join their church (if you’re not born into it), you have to go through RCIA, the Rite of Christian Initiation of Adults. It’s a nine-month-long process requiring weekly classes and weekly church services. You have to really want to do it to go through with it — this is no spur-of-the-moment commitment.

Sometime in the future I may discuss why I chose to convert, but for now I’ll simply say that I’m grateful to my generous, wonderful parents who supported me throughout the process, even as they must have wondered what kind of crazy thing their daughter was getting herself into.

The Easter Vigil was where I became confirmed as a Catholic. (I was already baptized a Christian in the Methodist Church, and the Catholic Church respected that.) I like to go back and revisit the Vigil, see my old friends from RCIA, and watch people become new Catholics. There’s something very moving about it; they’ve worked so hard for this moment, and they’re so excited to have made it. Some are on their second or third try, having dropped out of previous classes.

When Father Bill doused each catechumen three times with water in the name of the Father and of the Son and of the Holy Spirit, I could feel the joy radiating out from the baptismal font to the back of the church where I sat. Really. That’s a special feeling that more than makes up for three hours of cranky six-year-old, bless her little heart. (I think a Sunday of sleeping in, chocolate, treats from the Easter Bunny, and hunting for eggs with the neighbor kids made up for three hours of torture in her little mind. Her brother was perfectly cool throughout the Mass, as it was something “adult”, and therefore something he wanted to do. Plus he got all of the Sunday benefits his sister got.)

It tied in nicely that this month my online reading group is discussing St. Augustine’s Confessions, a work to which I alluded back in September. I alternated reading it and listening to excellent lectures from The Teaching Company about it this weekend while I worked out in the garden between family times.

I don’t have a deep conclusion to draw about adult baptism and whether or not Augustine was right. It’s a point to ponder. Would I now be better behaved if I had not received that sacrament until age 29? I don’t really know.

I can say this much, however: this Easter weekend, I am grateful to be with the people I am with, in the place that I am, in the life that God has given me. I pray that I will continue to feel gratitude, no matter my life’s circumstances. Amen.

Blessings to you and your family this Easter.

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Flatland

March 30, 2010

I’ve been thinking about how to finish up my post on the Fibonacci sequence, and what’s been holding me up is how to illustrate my point. I mean that literally. How am I going to draw a golden rectangle and the spiral that can be formed inside repeated rectangles that corresponds perfectly to certain shells in nature? I can’t find my compass, and I’ve lost my momentum.

So here’s a Fibonacci spiral from wikipedia:

The ratio of the larger side of the rectangle to the smaller is the same as the ratio of a larger Fibonacci number to one just preceding it, a number that approximates 1.61803.  It’s known as the golden ratio, and it’s been used since ancient times in art and architecture.

I’m stopping there because I have become completely absorbed in something else: the concept of dimensions.

This was a later lecture in my math series, and it touched on a subject that I’ve run across in various books by physicists like Brian Greene and Richard Feynman. Every time I read something about dimensions beyond the third, I get curious. The physicists think they’ve proved 17 dimensions now, but it’s all theoretical. Still, I like the idea.

One of the math professors in this lecture series began to illustrate the dimensions we have experience with. First we start with a point. A point is invisible, mathematical, and has no dimension. It is completely self-contained.

Next we have a line. A line stretches between two points. It moves only in a forward or backward direction. It is one-dimensional.

Next would be two perpendicular lines. Together they would form a plane. We could draw a square on the plane, and it would have four corners, or vertices. It would have four sides, all made of one-dimensional lines. The square is two-dimensional.

And finally we could draw a line perpendicular to the plane (to go up and down). We could turn the square into a cube. It would have eight corners, or vertices, and six sides made up of two-dimensional squares. The cube is three-dimensional.

If we kept up this pattern, a four-dimensional cube would have 16 vertices and eight sides made of three-dimensional cubes. To see the fourth dimension we’d need to be able to draw a line perpendicular to the third dimension. And of course we can’t do that, so this theoretical concept is one we can’t visualize.

You know how you can draw a cube, a three-dimensional object, in 2-D? Example:

You look at it, and the sides are not all equal and the angles aren’t all 90 degrees, but it doesn’t matter because we have experience with cubes and so are able to visualize that this drawing represents one.

Well, one of the professors constructed a 3-D model of a 4-D cube. It was sort of the same thing as the drawing above, only it was three-dimensional and had eight sides. He suggested that to someone in the fourth dimension it would make sense, like the drawing above makes sense to us. Click here for a movie of what it looks like.

Science fiction writers call the 4-D cube a tesseract. I first ran across that term as a kid reading A Wrinkle in Time. It’s still one of my favorite books.

The professors suggested that a better way to wrap our brains around the concept of dimensions is to go lower rather than higher. We live in 3-D. But what would be different about life if we lived in only two dimensions?

This is a concept Edwin A. Abbott explored in his groundbreaking 1884 novel, Flatland. The book is in the public domain now, and you can read it in its entirety here.

The main character is a square, and he describes his existence. Take a penny, for instance. You lay it on a table, and you can see that it’s round. But what happens if you put it on the edge of the table and lower your eye so that your view is in the same plane as the penny? It becomes a small, copper, horizontal line. In Flatland, the square explains, everything is on a plane, so everything to the creatures in Flatland appears to be a line. He goes into a bit of detail about how they learn to recognize people and places, and if you’re interested, you can check out the link.

This was a part that grabbed my interest: in Flatland, the surface of each figure is its outline. To a square then, the outline of a square is its skin. It is incapable of seeing anything contained within the outline without cutting itself open. So far as a square is concerned, the area within it is as contained as our organs are within our bodies. But to a three-dimensional creature, the insides are visible.

Below is an example I drew. On the left is a girl as we would expect to draw her in a 2-D space. On the right is a girl as Abbott might have drawn her. For her to see or hear, her eyes and ears would need to be on the perimeter (surface) of her body. Additionally, her insides, while invisible to her and her compatriots, would be visible to us.

So, to extrapolate this idea to the fourth dimension, fourth-dimensional creatures would be able to see our insides just as we can see those of two-dimensional creatures. Pretty freaky.

If you’re wondering why such a weird concept has grabbed my attention, the answer is that I’m reworking a time-travel novel I wrote, and I’ve been looking for a good way to explain how it works. So I’m kicking these thoughts around in my head. Perhaps time travel can be explained through other dimensions, blah, blah… I’m still figuring it out. But it’s fun for me.

Hope you enjoy it too!

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Everything is Golden

March 25, 2010

I like pineapples, despite their being a complete PITA* to cut up. They’re tart and tangy and a little bit sweet, which is a lovely combination of flavor. They’re even the secret ingredient in every episode of Psych, one of my favorite TV shows, which you can watch here.

But pineapples have their own little secret.

Have you ever noticed that the basket-weave look to a pineapple is just spirals that go in two different directions? Here’s a weird factoid: all pineapples have the same number of spirals. In one direction that number is 8. Guess what the number is in the other direction. Go on. Guess.

What’d you say? If you’re like me, you figured it’s somewhere between 7 and 9.

But it’s not. Its 13.

Don’t believe me? That’s okay. I was told this in my latest nerdy endeavor, which is to watch a series of lectures called The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas. And I didn’t believe it either.

So, Doubting Thomasina that I am, I went into the kitchen, pulled a whole pineapple out of the refrigerator**, and counted the spirals. Sure enough, there were 8 one way and 13 the other.

Turns out pinecones have 8 spirals one way and 5 the other. Cone flowers have 13 spirals in one direction and 21 in the other. And daisies have 21 spirals in one direction and 34 in the other. Sunflowers have 55 and 89.

Yadda, yadda, who cares, right? Well I do, because I’m nerdy that way.

You may notice that some of these numbers overlap. (Weirdly, so do some of the names, like pine/cone/flower. But that’s neither here nor there.) If you line them up you get 5, 8, 13, 21, 34, 55, 89. This makes a pattern whereby the first two numbers in the series add up to the third number, the second two add up to the fourth number, and so on. So you can extrapolate the series forward and backwards to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.

The guy who first noticed this phenomenon and put it all together into a sequence of numbers was Leonardo of Pisa. So in his honor it’s named the Fibonacci Sequence.

Yeah, I don’t get it either.

But Leonardo did this in 1202, when he published a mathematical paper about rabbits propagating. His question was — how many rabbits would you get if you stuck a pair of baby rabbits in a fenced-in area for a year?

Here are the rules: 1) rabbits take two months to mature, 2) after those first two months they will produce a pair of rabbits every month, and 3) we won’t worry about real-life concerns like feeding all these rabbits, cleaning up after all these rabbits, smelling all these rabbits, or freaking out that all of these rabbits are blood relatives and probably genetic mutants.

In January, we’ve got 1 pair of baby rabbits. In February, they reach maturity and we’ve still got 1 pair of rabbits. March, we’ve got a pair of baby rabbits and a pair of adult rabbits, making 2 pair. April, it’s two pairs of adults, 1 pair of babies, for 3 pair. Then it’s 5 pair, 8 pair, 13 pair, on up the Fibonacci sequence until December, when we hit 233 pairs. You can see this idea demonstrated (in a G-rated way) here.

It’s looking like this sequence is repeated in nature quite a bit already, but it goes even farther. And it gets more interesting. I promise.

But I’m tired, so I need to call it a night. More to come later.

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*Pain In The Ass

**On a recent visit to a Costa Rican pineapple plantation, my parents learned many facts about how to choose a good pineapple. Among other things, they were told unequivocally to refrigerate the pineapple as soon as they brought it home from the store. So now I do that too.