Math For Grownups

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?

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My birthday is this week, and I’ve decided I don’t want to work so hard. So today, I’m bringing you three archived Math at Work Monday interviews–two artists and a museum curator.  Enjoy!

Blossom, layering of enamel over silver. Photo credit: Hap Sakwa.

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.

drawer #4 from Marcum's "collections" series

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.

An installation at the BMA.

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!

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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.

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