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Math and Makeup

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 
can't find anything. Will you please help?
Ashley Peterson

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   

   Balancing Chemical Equations   

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   

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 
that comes from your brain's own knowledge about your body, but when 
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 

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:   

I hope this helps. If you have other questions about this, please 
write back.

- Doctor Achilles, The Math Forum   

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?

Hope this helps.  If you have other questions about this or you're 
still stuck, please write back.

- Doctor Achilles, The Math Forum   
Associated Topics:
High School Physics/Chemistry

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