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### Math and Makeup

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Date: 01/05/2002 at 00:15:01
From: Ashley Peterson
Subject: How math relates to makeup!

Dear Dr. Math,

My teacher told us that everything relates to math. Well, I was
wondering how makeup relates to math. It's something I use every day
and I thought it would be cool to know. Plus I might even be able to
impress my teacher if I knew how. I've looked on the Internet and

Sincerely,
Ashley Peterson
```

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Date: 01/05/2002 at 00:42:00
From: Doctor Achilles
Subject: Re: How math relates to makeup!

Hi Ashley,

Thanks for writing to Dr. Math.

Two (pretty much unrelated) ideas come to mind. Both of them have to
do with science:

The first is that makeup itself is either a complicated synthetic
material or some natural material that's been purified.

In the first case (the complicated synthetic) it takes a lot of math
to engineer it properly. Making synthetic chemicals is a lot like
cooking, only you use huge batches of ingredients and you have to
measure them out precisely and mix them under controlled conditions.
Figuring out how to make new synthetic materials that no one has ever
made before requires you to do a lot of chemistry. Even simple
chemistry is a lot like algebra; you have to work with equations. And
if you're really serious about chemistry, you need to have a fair
amount of calculus.

In the other case (the purified natural material) you have to do a lot
of the same kinds of math only you're not mixing things, you're
unmixing them. Figuring out how to get one chemical out of a mixture
requires an even better understanding of chemistry and the math that
goes into it than making chemicals. (Although, most times, when you
make a chemical you have to purify it from by-products.)

For a look at how math and chemistry relate, check out:

Proportions for a Chemical Solution
http://mathforum.org/dr.math/problems/fair.6.24.96.html

Balancing Chemical Equations
http://mathforum.org/dr.math/problems/katie2.11.21.99.html

The second idea is about how you actually apply makeup. This requires
your brain to do a lot of math subconsciously. What you are presented
with is a view from two points on your face (your eyes) of an inverted
reflection of your face. Just figuring out where your real face is in
two dimensions from that requires knowledge of the math. For a look at
some of the math involved in mirrors, check out:

Calculating a Mirror's Reflection
http://mathforum.org/dr.math/problems/brandon.8.8.99.html

But you don't just know from that where all the parts of your face are
left-right and up-down; you also know how far back they are. A lot of
you're putting makeup on your eye, it's probably a good idea to double
check in a mirror so you don't miss and blind yourself. Your brain
does some really complicated computations on the slight difference
between how an image looks in each eye to figure out how far back it
is. The greater the difference, the closer the image.

And all of that math just tells your brain where your face is relative
to your EYES. But you need to put makeup on with your hands. So then
you have to use more data from the muscles in your arm and hands (as
well as data from the mirror image of your hands) to figure out where
your hands are relative to all of this so you can apply the makeup
properly.

The fact that you're probably not holding still at all throughout this
makes the math ten or twenty times more complicated. You have to infer
from where your hand is and how quickly it is moving now where it will
be in a few hundredths of a second and how fast you should move it,
and all the while your face might be moving left or right.

In fact, the amount of math you have to do subconsciously every time
you put on makeup would probably take your desktop computer weeks, and
no artificial computer on the planet could even come close to your
performance. And you do it all without even thinking about how
complicated a job you're doing. In fact, most of the time you're
probably thinking about something else entirely.

I found all the pages I suggested to you in the physics and chemistry
area of the Dr. Math archives, located here:

http://mathforum.org/dr.math/tocs/physics.high.html

write back.

- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/
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Date: 01/05/2002 at 13:26:04
From: Ashley Peterson
Subject: How math relates to makeup!

Thank you very much for your help. I would never have figured that
out on my own.

I have one last question: I would also like to know how all of this
relates to algebra.
```

```
Date: 01/05/2002 at 21:07:19
From: Doctor Achilles
Subject: Re: How math relates to makeup!

Hi again Ashley,

Thanks for writing back to Dr. Math.

The best link I see is with the chemistry. If you check out the
chemistry links in the last message, that'll give you a start.

As I said, the basic idea of chemistry is a lot like cooking.

Let's say that I know that 100 grams of makeup ingredient A plus
50 grams of makeup ingedient B will make me 80 grams of makeup and
70 grams of garbage by-product that I don't want.

Let's say that I can remove 90% of the garbage by-product using
chemical X, but that also removes 10% of the good makeup product.

And let's say that 500 grams of the final makeup fills a volume of
1 liter.

So if I'm making makeup and I want 2 liters of 90% pure makeup, how
much ingredient A should I use, how much ingredient B?

- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Physics/Chemistry

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