Hair Maths 1: The curly problem with curly hair

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More about Troy’s $million hair here. Hair statistics including how many hairs a human has on their head here.

Mathspig studied hair chemistry at uni. Tricky stuff. Put simply, hair is made of long strands of protein called keratin held together by sulphur (and some hydrogen) bonds. To curl hair, the keratin strands in the outer curve of each hair has to be stretched with curling tongs or hair curlers, heated and dried. The bonds in each hair reform with one side longer than the other … Hence, the hair curls like gift-wrap ribbon. But high humidity allows hair to reabsorb water and straightened hair just goes psycho curly again!

This excellent hair diagram comes from The Chemistry of Shampoo and Conditioner, in an article by EMMA Dux for the Royal Australian Chemical Institute

Some people are born with hair follicles that produce keratin at different rates across the follicle. They have curly hair. Hair perms chemically break and reform the sulphur bonds while the hair is held in small curlers (curly hair) or a very big curlers(relatively straight hair.) thus permanently curling the hair.

Here’s the Maths:

Curly hair looks like a 3D Helix.

More on 3D helix maths here

But, in fact, one strand of curled hair looks more like a spiral staircase.

The outer edge of the staircase is longer than the inner edge.

 

 More helix maths here.

CHALLENGE:

Mathspig doesn’t expect Middle School students to plot a 3D Helix. But if they have started TRIGONOMETRY then they can see that the maths they are studying is used in CGIs for films and computer games in this case to generate realistic curly hair!!!! That’s cool. This maths was needed to model Merida’s curly hair in  BRAVE.

SMARTY PANTS CHALLENGE:

Some middle school students could calculate some points on the helix.

Now students must be introduced to radians.

Simple EXPLANATION: Angles eg. 30⁰ are not useful in calculations but fractions are very useful.

Eg. The circumference of a circle:

C = 2πr

Now imagine if you scan with a floodlight set at a radius of 1 km. So:

C = 2π

So the circumference is 2π.

You scan ¼ of a circle, the distance the light moves is ¼(2π)

or ½ π or 1.57 km (see below)

This measurement of an angle is in RADIANS.

0⁰     = 0 circle = 0

45⁰   = 1/8 circle = 2π/ 8 = 2 (3.14) /8 = 0. 79

90⁰   = 1/4 circle = ½ π = ½ (3.14) = 1.57

135⁰   = 3/8 circle = ¾ π = ¾ (3.14) = 2.36

180⁰= 1/2 circle = π = 3.14

225⁰   = 5/8 circle = 5/4 π = 5/4(3.14) = 3.93

270⁰ = 3/4 circle = 3/2 π = 4.71

315⁰   = 7/8 circle = 7/4 π = 7/4 (3.14) = 5.50

360⁰ = 1 circle = 2π = 6.28

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You will find Cos  tables at NASA Sine tables at Mathhelp

 

Answer here: Answers- 3D Helix Table

 Advanced students may want to look at what the Uber Geek 3D Helix generating program at the free graph website PLOTLY here.

Hair Maths 2: Why CGI needs maths

 The challenge is to make CGI hair look real. This isn’t easy. Some CGI hair doesn’t need maths because it doesn’t move. Some CGI is just ugly. Eg.

More from the worst video game hair cuts ever here and here.

In the world of CGI hair, curly hair is the challenge. It is difficult to model.

According to Pixar animators:

Hair curls due to the way it is grown. Curly hair is almost like a ribbon, while straight hair is more tubular.

(Mike Seymour,Brave New Hair Fxguide)

 You will find detailed maths used by Pixar to model Merida’s hair here.

So straight hair swishes and curly hair springs or bobs.

Ariel, the Little Mermaid, was meant to have curly hair, but animators stuck with a ‘flowing block’ of hair. Before Ariel Disney Princess often had up-dos.

Progress was made with Merida’s hair in the Disney/Pixar animation BRAVE. See BRAVE NEW HAIR WIRED and fxguide.

According to Rachel Gross of Wired Magazine in 2009 Chung’s team designed a new simulator named Taz, after the wild Looney Tunes character. It forms individual coils around computer-generated cylinders of varying lengths and diameters. The resulting locks stretch out when Merida runs but snap back into place as soon as she stops. Add a little randomness, some gravity, and more than 1,500 hand-placed corkscrews and flyaway wisps and voilà: hair with depth and texture viewers have never seen before. The result may look wild, but it’s not. “It’s very stylized, very controlled,” Chung says. No hair spray required.

Rachel Newar explained last year in the Scientific American that physicists have modeled the movement of a single curly hair.

But, co-author Pedro Reis, an assistant professor of mechanical engineering and of civil and environmental engineering at MIT noted. “ the geometry of a curly hair is highly nonlinear— a word we often use for something complicated.”

The model could also calculate curvature of steel pipes or other spooled material. “We were engineers trying to solve practical, useful problems from the start,” Reis says. “I’m not a professional hairstylist—I’m bald, actually.”

More @ Mashable Video here.

And, from Stanford, if you’ve ever been curious this is what a curly hair algorithm looks like ie. It is a computer program for curling graphic hair.

So you don’t have to eat your crusts after all.

What were mothers thinking!!!!

Hair Maths 3: How OLD is your hair?

The follicle diagram (above) came from bgoffforensics blog. More on hair growth rates here.

QUESTION 1:

QUESTION 2: 

The growth rates of 1.25 cm/month or 0.5 inches/month are rounded off. Calculations using these growth rates will produce different answers for very large numbers (as above). Read more about the above story in the Daily Mail, UK here. By the way, Asha has been growing her hair for 25 years. Her hair weighs 3 stone or 19 kg. You can find more hair records including the World’s longest ear hair here.

QUESTION 3:

Natasha lives in Rio de Janeiro, Brazil. Read more about Natasha in the Daily Mail here. Once again Mathspig has rounded off the length of hair for easy calculation. The Daily Mail puts Natasha’s ‘mane’ at 5 feet 2 inches or 157 cm. This is the length of hair to be cut off. So Mathspig adjusted the length to 180cm or the approx length of her longest hair.

Hair Maths 4: The Great SHAMPOO rip off

According to Salon.com:

There are two types of ingredients in shampoo. One type cleans your hair. The other type strokes your emotions.

Shampoo has one main ingredient:

DETERGENT

The two most common shampoo detergents are ammonium lauryl sulphate and ammonium laureth sulphate. That’s all you need to clean your hair. All those shampoo bottles. All that supermarket shelf space offer THE SAME PRODUCT.

Other ingredients include: perfume, dye, preservative, foaming agent, acid (The problem is hair swells in an alkaline solution eg with soap when rinsed out the hair shrinks making knots impossible to brush out. ARggghhhhh!), thickeners, thinners, sheen additive, pearlesence powder (makes shampoo look pearly) etc

According to Steve Antczak, Co-author of Cosmetics Unmasked: Your family guide to safe cosmetics and allergy-free toiletries,

A typical shampoo is mostly water, containing between 5 and 20 per cent detergent, with shampoo for dry hair containing less detergent than shampoo for greasy hair.

See New Scientist, Sept , 2009

So the potency of shampoo looks something like this: 

Assume you buy a 500ml bottle of shampoo for $10. Now you can work out how much some shampoo manufacturers rip you off by filling in this table:

 

How much do you pay for water in shampoo? The Answer is here. Mathspig % detergent in shampoo ANS.  

What does Bill Bunn from Salon.com recommend?

My new shampoo, Sunlight Dish Detergent, has just four ingredients. It’s runny and slightly acidic, smells vaguely lemony, doesn’t foam excessively and looks anemic.