Photo courtesy of potzuyoko

In our interview on Monday, professional photographer Sally Wiener Grotta talked about using histograms to help determine the exposure she needs to best reflect her subject in a photograph. If you took any statistics in high school or college — or have helped a middle schooler with her math homework — you may know exactly what a histogram is. But do you understand how these graphs are helpful for photography?

In short, a histogram is a graph that demonstrates variance and frequency.  (Stay with me here. I know there are some strange, mathy words in there.) Here’s a really simple example:

The administrators of a health clinic are collecting data about the patients, so that they can provide the most appropriate services.  The histogram below shows the ages of the patients.

Even with one quick glance, it’s apparent that the clinic sees far fewer patients who are between 80 and 90 years old. In fact, it looks like the group that’s most represented includes those between 40 and 50 years old.

(If you’re really being a smarty pants, you might notice that the histogram follows the normal or bell curve. But you don’t have to know that to get along in everyday life — unless you work in statistical analysis.)

So here’s what’s special about a histogram:

1. The horizontal line (or axis) represents the categories (or bins). These are almost always numbers, and each one has no gaps. In other words, in a histogram, you won’t have categorical data, like people’s names. Notice also that the data is continuous. Someone who is 43 and 5 months falls in the 40-50 year old category.

2. The vertical line (or axis) represents the frequency or count of each category.  These are always numbers. So in the histogram above, 40 people who visited the clinic were between 80 and 90 years old.

3.  The bars of the histogram butt up against one another. That demonstrates the fact that there are no gaps in the data and the data is numerical.

4. The taller the bar, the more values there are in that category. The shorter the bar, the fewer values there are in that category.

So let’s look at a photographer’s histogram:

First off, these histograms are automatically generated by imaging software or even some fancy-schmancy cameras. In other words, technology plots these values. It’s the photographer’s job to interpret them.

You probably noticed that there are no numbers on this histogram.  Like a statistical histogram, the vertical axis represents frequency.  But the horizontal axis doesn’t represent numbers. Instead, it shows shades.  Follow the bar at the bottom of the histogram from the left to the right.  Notice how it goes from black to grey to white? In fact, the bar gradually changes from black to white.

If you could blow up this histogram to a much larger size, you would see that it’s made up of lots and lots of skinny rectangles. These represent the number of pixels in the photograph that are each shade. So there are very few (if any) pure white pixels. There are some pure black pixels, but not as many as there are grey ones.

By glancing at this image, an experienced photographer can determine whether an image needs more or less exposure. There’s a great deal of artistry in this — a really dark photo can have a dramatic effect, while certain conditions require more exposure than others.

There you have it. Histograms aren’t just for statisticians. And those silly little graphs you drew in your middle school math class actually have artistic value!

Do you have questions about histograms? Ask them in the comments section!

Photography is one of those art forms that looks easy but is really challenging — at least challenging to get it done right!  Writer and photojournalist, Sally Wiener Grotta describes how math helps her compose the best photograph, including perfect lighting. 

Can you explain what you do for a living?

Essentially, I am a visual and verbal storyteller. This has developed into a multi-pronged career.

As a photojournalist, I have traveled all over the globe, visiting all 7 continents (including Antarctica several times) and many islands (such as Papua New Guinea and Madagascar) on assignment for major magazines and other publications. My current and ongoing fine art project is American Hands (www.facebook.com/AmericanHands) for which I am creating narrative portraits of individuals who are keeping the old trades alive, such as a blacksmith, glassblower, bookbinder, spinner, weaver, etc.  I travel around the county, mounting American Hands exhibits and giving presentations about the people I photograph.

In addition, I give lectures and teach master classes on photography and imaging. I recently launched a YouTube channel in which fellow photographer David Saffir and I discuss the essential elements that define a photograph and pull us into it, using the narrative power of shadow and light.

As a non-fiction writer, I have written literally thousands of articles, columns, features and reviews for major magazines, newspapers and websites, as well as seven non-fiction books. In non-fiction, I am primarily known for my expertise in testing, analyzing and explaining technology related to photography, imaging, printing and epublishing.

My first novel “Jo Joe” will be published this spring as both an eBook and printed book by Pixel Hall Press, followed later this year with other stories and books.

When do you use basic math in your job?

Math is integral to my work in many ways. An intuitive understanding of geometry is essential for good photographic composition. In addition, I use math to control exposure (the amount of light used to define a photograph) and to decide how to set up auxiliary lighting.

This is a histogram that displays exposure, contrast and saturation of one of my own photographs.

A prime example of math in photography and imaging is the histogram tool. The histogram is a graph that provides information analyzing the exposure of a photograph. When a photographer or digital artist looks at a histogram, it helps us understand the “dynamic range” of the picture. In other words, what percentage of the photograph is made up of highlights, shadows and midtones. If the graph displays that there is too much image data in, say, the highlights, and I know that the image is of a scene that isn’t that bright, I can then decide to change my exposure so the photo better represents the scene.

But basic math goes much deeper into my everyday career concerns. For instance, my American Hands project is a non-profit venture supported by grants and sponsors. When I apply for a grant, I must present an accurate, logical and meaningful balanced budget. Therefore, I have to calculate my costs over time and balance that against potential income. (If the budget isn’t balanced with income=costs, the grant application will be rejected.)

Richard Moore, broom maker, from American Hands.

Another example of everyday math has to do with laying out books and journals for publication, such as my American Hands Journal. At the very basic, a typical book is printed in “signatures” of a specific number of pages each, such as 4-pages each. So a book must be laid out so that its total pages are a multiple of 4 (or whatever the signature number is). Then, there are spatial concerns, such as keeping type and photographs within specific printable margins, that requires more intuitive understanding of geometry.

Do you use any technology to help with this math?

I do believe that it is important to understand math and be able to do it without calculators or computers. However, when I use it for accounting, grants applications and such, I must be sure that I haven’t introduced an error, through a mistake in arithmetic or simply a typo. So, I may use a calculator. More often, I will use Microsoft Excel on my computer to create a spreadsheet that does automatic calculations for me when I input figures. However, I am the one who creates the rules for those calculations. So, using a spreadsheet doesn’t preclude the need to understand the underlying math.

How do you think math helps you do your job better?

Math isn’t only necessary in my career as an artist and writer, but it is also a skill that sharpens your mind the more you use it. That kind of precision thinking is a great complement to the creative side of my business, balancing it. What’s more, a sharpened mind is one that is more open and creative.

How comfortable with math do you feel?

I was lucky to have some wonderful teachers – starting with my mother before I ever went to school. She created basic arithmetic puzzles to keep me busy, and I learned to think of numbers as a game, starting when I was about 4 years old. So, I have long been comfortable with numbers and their relationships to each other. Math and art are not opposites. In fact, in the Renaissance, the great mathematicians were artists and vice versa. And, today, the great math innovators have highly creative minds.

What kind of math did you take in high school?

I studied geometry, algebra and calculus in high school.  I enjoyed it, again, mostly because I had good teachers. It continued to be a game to me to understand how numbers fit and changed each other.

Did you have to learn new skills in order to do the math you use in your job?

The new skills I developed since leaving school has to do with defining intelligent, useful calculation rules in an Excel spreadsheet. But it was all based on math I already understood, so it was relatively easy… once I understood how the spreadsheet works.

Do you have questions for Sally?  Ask them in the comments section!

Picasso's Violin and Grapes (Photo courtesy of ahisgett)

When my brother Graham was in kindergarten, he learned a little bit about Pablo Picasso.  And so my mother decided to take the whole family to a touring Picasso exhibit at the Smithsonian, which featured five or so of his paintings, including some of his most famous examples of cubism.

My brother is a man of few words, and he wasn’t any different as a little boy.  He quietly walked around the paintings, looking intently at them and being careful not to cross the red velvet ropes that kept out curious hands.  Nearby, we were all watching Graham, wondering what in the world he was thinking.

That’s when he stepped back from one of the paintings and said, “Oh, I get it.” We waited for something insightful. He pointed to the velvet ropes and said: “The paint is still wet.”

Cubism is not the easiest kind of art to understand.  But you have to admit — whether you like it or not — cubism catches the eye.

In cubism, objects are deconstructed, analyzed and reassembled — but not necessarily in their original order or size. When this is done in painting, the result is a three-dimensional object reassembled in a two-dimensional space, without regard to what can actually be seen in the real world.  So while you can’t see the back of a violin when you’re looking at the front, Picasso may depict the back and front at the same time in the same two-dimensional space.

Freaky, right?

I’ll leave it to the art experts to explain why this works.  But I can talk a bit about the

Portrait of Pablo Picasso by Juan Gris (Photo courtesy of Raxenne)

geometry of cubism.

First, you need to know that cubism has its roots in the work of Paul Cezanne. He began playing with realism, saying he wanted to ”treat nature by the cylinder, the sphere, the cone.”  In other words, he began replicating these figures as he saw them in his subjects.

Henri Matisse, Picasso and others took Cezanne’s approach even further.  It’s not hard to recognize the cubes and angles and spheres and cones.  But it’s the flattening of three-dimensional space and disregard of symmetry that really distinguishes cubism from realism or impressionism.

Symmetry is a very common occurrence in mathematics.  From symmetric shapes to the symmetry of an equation (remember: what you do to one side of an equation, you must do to the other!), it’s fair to say that when symmetry is absent, it’s a big deal.

And the same is true for nature, the most often referenced subjects in art.  A face, a water lilly, the body, a beetle — you could spend all day finding symmetry in the natural world.  Cubism turns this notion on its head.

And still the pieces are compelling.  It’s that dissonance that draws our attention and even illustrates difficult subjects. (Picasso’s most enduring and controversial pieces is Guernica, a large painting depicting the Nazi bombing of a small Spanish town.)  The artists do this by breaking traditional rules and ignoring some mathematical truths.

Do you like cubism? Have a favorite artist? When you’ve seen cubism in the past, did you think of it mathematically?

Sketch courtesy of anyjazz65.

My middle school daughter aspires to be a fashion designer, and so she’s been concentrating lately on learning to draw female human figures.  Last Friday, she came home from school and immediately logged on to the internet in search of a “how to draw” tutorial.  She spent the next several hours engrossed in a YouTube video that not only demonstrated how to draw the ideal human figure but offered some interesting tricks of the trade.  For example:

1. The ideal figure is eight heads tall.
2. The width of this figure’s shoulders is typically two heads — arranged horizontally — wide.
3. The width of this figure’s hips is typically two heads — arranged vertically — wide.
4. The top of this figure’s inseam (or the “bend” of the figure) is four heads tall or half a person’s height.

That’s right! Your own body can be sketched based on the size of your head!

What does it have to do with math?  This approach to drawing is based on proportions, and it depends on a relative unit.  In other words, the entire figure can be drawn based on one relative measurement — the size of the figure’s head.

(Here’s an interesting video that shows how to draw these figures by first folding the page in half longways and then in eighths along the short side. Great use of proportions!)

This approach allows great flexibility.  For example, men are typically taller than women, but their heads are also typically larger.  Therefore, the unit for a male figure will probably be bigger than a unit for a female figure.

In addition, artists can use this one unit to draw figures of varying sizes — tiny in one drawing or huge in a large-scale piece — simply based on this one unit.  All they need to do is draw the head first.

This photograph demonstrates foreshortening. Notice how the angle of the shot makes the feet seem much larger than the head. (Photo courtesy of hunnnterrr.)

It’s important to note that no one has a perfectly proportioned body.  Some people may be only 7.5 heads tall.  Or perhaps their legs are not half their height. Or maybe they have a long waist.  And the angle at which a figure is positioned will affect these proportions.  Objects that are closer seem larger, while objects that are farther away seem smaller. This is called foreshortening.

And of course anything can be used as the unit measure.  Have you ever seen an artist look at her subject over an outstretched brush or pencil?  This is a common method of measuring the figure from that particular angle.  An artist using the photograph to the left might notice, for example, that the subject’s right foot is three heads high.

The pencil or brush can also be useful in determining angles.  Two pencils can be held up to form the angle made by the figure’s arm and torso and then checked against that angle in the drawing.

All of these techniques are based on the properties of similar figures.  If two figures are similar, they have the same shape, but are proportional in size.  Remember your geometry class, when you proved that two triangles were similar, using the SSS, SAS and ASA similarity theorems for triangles?  (If not, don’t worry.)  They boil down to one important fact: all of the corresponding sides of similar figures are proportional, while all of the angles of those figures are the same measure.

But here’s the thing: artists probably don’t think too much about that.  My daughter hasn’t even studied similarity yet, but she’s able to figure out how to draw a human figure.  Once again, we’re using math without knowing the reasons behind it.  And that’s okay. It’s enough to know that it’s there.

Do you draw?  Have you attempted to learn to draw but not understood how to get the proportions right? Does having some of these rules help?

Shana Kroiz: Jewelry artist

Shana has been designing museum-quality jewelry for almost 20 years.  She also began the Maryland Institute College of Art (MICA) jewelry department, when she was fresh out of college.  Using a combination of resin molds, metals and gemstones, her pieces are distinctive and tell interesting stories.

Ursula Marcum: Glass artist

Ursula isn’t a glass blower, like Elizabeth Perkins.  Instead she works in kilnformed glass, creating layered pieces that truly unique. She uses various formulas to create her pieces, allowing different kinds of glass to fire at different temperatures and for different lengths of time.

Ann Shafer: Museum curator

Ann is the associate curator of the prints, drawings and photographs department at the Baltimore Museum of Art.  Part of her job is acquisitions, so she helps manage a budget — making sure that the museum has a balanced collection and spends its donations wisely.

Enjoy the interviews. See you on Wednesday!

Photo courtesy of Our United Villages

Today, I bring you an excerpt from my book, Math for Grownups.  Enjoy!  (Now you have your weekend project planned for you.)

Hanging pictures can be a tricky business. If you’re not careful, your foyer can look like a hall of mirrors—with crooked photos of your wedding party alongside drawings that your kid made in kindergarten. Not to mention the holes in the drywall from when you realized that you hung your college diploma so high up the wall that only a giant could read it.

Not exactly the look you were going for?

You may not want to face it, but a tape measure, pencil, and yes, even a level, are your best buddies in home decorating. And hanging anything on your walls is no exception. Let’s look at this in a bit more detail.

Mimsy Mimsiton is thrilled to have finally received the oil portrait of her dear Mr. Cuddles, a teacup poodle who is set to inherit her large fortune. The painting will look fabulous above the marble fireplace in the west-wing lounge of her mansion.

But drat! The museum curator Mimsy has on retainer is in Paris, looking for additions to Mimsy’s collection of French landscapes. (She’s redoing the upstairs powder room and wants just the right Monet to round out the décor.)

But the painting must be hung before Mr. Cuddles’s birthday party. His little poodle friends would be so disappointed not to see it! There’s no way around it; Mimsy’s poor, overworked House Manager must hang the painting herself.

Luckily, House Manager is no stranger to the DIY trend, and Butler will be there to help. The two meet in the lounge, where the painting has already been delivered—along with a stepladder, a tape measure, and a pencil. Once House Manager marks the spot, Handy Man will come along to safely secure the painting to the wall.

House Manager and Butler get to work. First they measure the painting: With the gilded frame, it’s 54″ tall and 60″ wide.

Next, they turn their attention to the space above the mantle. House Manager climbs atop the ladder, while Butler holds it steady. From the ceiling to the top of the mantle is 84″.  The width of the mantle is 75″.

Climbing down from the ladder, House Manager notes that the painting will certainly fit in the space allotted. She knows from experience that it is to be centered over the mantle. However, Mimsy will have a fit if the painting is centered vertically—between the ceiling and the mantle. No, the bottom of the painting must be exactly 12″ above the mantle.

So how high should Handy Man install the picture hanger?

To find out, House Manager must add 12″ to 54″ (the height of the painting). The top of the painting should be 66″ above the mantle.

House Manager grabs her tape measure again and removes the freshly sharpened pencil from behind her ear. Then she climbs the ladder. Starting at one end of the mantle, she measures 37½”—which is half the width of the mantle. She makes a barely visible pencil mark at that point.

Then from there, she measures up the wall to 64″. Again, she carefully makes a faint pencil mark.

If House Manager stopped here—leaving that small mark for Handy Man to hang the portrait—she’d probably be out of a job. That’s because she’s merely marked the top of the frame, not where the hanger should be secured.

She descends the ladder and goes back to the portrait. Turning it around, she notices the picture wire that has been stretched from one side to the other. She hooks her finger under the center of the wire and pulls up gently—creating an angle, as if the picture wire were hanging on a nail. Now an angle is a two-dimensional figure formed by two lines (called rays) that share a common point. Hereʼs an easier way to remember this: An angle looks like a V.

If she can measure the distance from the top of the frame to the vertex—the point where two sides of an angle meet—she’ll be in business.

There’s just one more thing to consider: Is the vertex of the angle too far to the left or too far to the right?  For the painting to hang straight and be centered on the mantle, the vertex must be located at exactly half the width of the portrait.

House Manager uses her tape measure to find the length of each leg of the angle. In other words, she measures the distance from one end of the picture wire to the vertex of the angle and then the distance from the vertex of the angle to the other end of the wire. If the vertex is centered properly, the legs of the angle will have the same length.

Moving her finger ever so slightly, House Manager centers the vertex of the wire angle—and measures from that point to the top of the picture frame: 9″.

She now can make the final mark for Handy Man. She climbs the ladder for the third time and measures 9″ from the mark she made earlier. Again, being very careful, she makes a tiny mark on the wall.

House Manager’s work is done. If anything goes wrong now, it’s Handy Man’s fault.

She folds up the ladder and gathers her supplies. Then she’s off to order beef cupcakes for Mr. Cuddles’s party.

Any questions?  Ask them in the comments section.

Turns out the Rule of Thirds isn’t really about thirds, per se.  Instead it’s about ninths.  The idea is to divide the image into nine equal parts — something like this:

Samantha Hand, plein air painting

Sam doesn’t remember this, but when she and I were in middle school, I used to ride home with her on the bus after school, when we’d watch Godzilla on television and eat her mother’s homemade potato bread.  At that time, she said she wanted to be a veterinarian (like her dad).  Instead she earned a BA in art and then her MFA. Since 2010, she’s discovered her talent in oil painting.  

Samantha Hand has some mad skills when it comes to oils and canvas. And even I was surprised by the math that she uses.  Unlike most of the other folks I’ve interviewed for Math at Work Monday features, Sam really counts on being able to visualize the math she needs.  Read on…

Can you explain what you do for a living?  For the last two years, I have completely immersed myself in oil painting and have tackled landscape, plein air, still life and portraiture. Currently I am painting compositions that intrigue me in hopes of selling them, while accepting commissions on a variety of subjects. Recent projects include still life and figurative painting.

When do you use basic math in your job?  I use the most math at the beginning stages of a painting. When I am sketching thumbnail ideas, I use the rule of thirds to compose a more interesting picture. I use a variety of angles to draw the eye toward the focus of the picture and to lead the eye around the composition. I also use angles in drawing perspective when I am attempting to create depth in a two-dimensional space. (For example, the angle of a building is wider in the foreground and will go toward a vanishing point as the building retreats into the distance)

If the composition is complex, as in a triple portraitI am currently working on, I use a grid to enlarge smaller reference images to the larger size of the canvas. This helps to keep the proportions of the sketch on the canvas accurate. Proportions are also important in balancing the values and subject matter in a composition. I check to see if the proportion of dark values is greater or lesser than the proportion of light values to add interest.  I may balance the visual weight of the subject with a greater space of sky to create visual tension or to draw the eye toward the subject.

When I am sketching the figure I am constantly checking my proportions by comparing the size of body parts. For example, in most faces the space between the eyes is the width of one of the eyes in the face. Also, in general, people are approximately 6 and a half heads tall. I use a paint brush or pencil to measure and compare. I also use this measuring and comparing in all other subject compositions to check my spacing and proportions.

Once I begin painting, I use ratios in the mixing of colors. If I am looking for a purple I may mix an equal amount of red and blue. But if I want a warmer purple with a reddish tint I’ll use less blue in the mixture. Throwing in the amount of yellow equal to the red will turn it toward a brown. Equal measures of red, blue, yellow becomes a neutral gray. There are infinite numbers of colors to be mixed which is one of the most exciting things about painting.

Do you use any technology to help with this math?  I do not use a calculator or computer because the math I use is simple and not very exact. It is more about the feeling of balance or rightness. If something doesn’t feel right with the composition I begin to check using more exact measurements and angles.

How do you think math helps you do your job better? Math is the building block of my compositions. I use angles and proportions to try and create intriguing compositions with believable subject matter.

How comfortable with math do you feel?  I am very comfortable with the math I use in my artwork but less so with the everyday math of a household. Somehow I feel as if I can visual the math I use in compositions and it makes sense to me. When I apply it to household tasks I have to really focus on the task at hand.

What kind of math did you take in high school?  I only vaguely remember my classes in high school but did take math analysis, geometry and the other algebra courses offered. I really enjoyed my math classes and felt confident in my ability, though less so with geometry. I continued with a calculus course in the first year of college and enjoyed that also. Unfortunately, I think I’ve only retained very simple math skills.

Did you have to learn new skills in order to do the math you use in your job?  I haven’t had to learn any new skills yet but I have learned to use the math I know in tangible situations.

Did you have any idea about the math that goes into planning a painting? If you have a question for Samantha, ask it in the comments section.

This month, Math for Grownups has gone arty, taking a close look at how math shows up in the visual arts.  Last week, glass blowing took center stage.

First off, there are the tools.  The steel pipe that holds the glass is a very long cylinder or straw.  The hole allows the artist to blow air into the glass at one end, which creates the bubble.

Then there are not one, not two, but three furnaces.  Why three?  Because the entire process requires different levels of heat.  The first furnace contains molten glass.  The second, called the “glory hole” is used to reheat the piece as it’s being formed.  And the third, which is called the “lehr” or “annealer” is used to cool the piece very slowly and deliberately so it maintains structural soundness.

The artist is constantly working against temperature changes.  When the glass is in liquid or semi-solid state, its shape can be changed, and this is accomplished by spinning the pipe. To achieve a symmetric shape, the glass must be spun in consistent circles.  This is where the bench comes in.  The glass blower can place the pipe along two parallel arms and push the pipe out and in.  Because the arms are parallel and the same height from the floor, the glass can be spun consistently.

Okay, so we have some geometry (the pipe and the bench) and measurement (the furnaces regulated at different temperatures).

Time for more geometry.  After the glass blower gathers a layer of glass on the end of her pipe from the first furnace, she rolls it on a table to give it a cylindrical shape.  Blowing into the pipe creates the bubble — which eventually will become the curve of a bowl, glass, lampshade or something altogether different.  How that bubble is formed is critical to the stability of the piece.  The glass must be thicker around the bottom and thinner along the sides.

And this is where things get really mathy.  See, the bubble at the end of a glass blower’s pipe is usually some kind of ellipsoid.  You already know what an ellipsoid is.  You live on one: planet Earth. An ellipsoid is like a slightly flattened sphere.  In fact, a sphere is a special kind of an ellipsoid.

After the glass blower completes the piece, it goes into the annealer, which is programmed for that particular piece of glass.  Some pieces need to cool more slowly than others, and that cooling process is dictated by math.

So there you have it — my very uneducated look at the math of glass blowing.  You too can see math in everything, if you just look closely enough.

Are you noticing math in art? Share your observations in the comments section.

Photo courtesy of Chibijosh

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.

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.