How "Phi" Is My Face?
Student Page

Introduction: We are going to work with photographs, make some measurements and see which face has the most perfect proportions. You may be thinking, how do I know if a face has perfect proportions? In the art world, if a face has certain measurements that are close to the golden ratio they are pleasing.

Let's look at these two examples:



Top of head to chin
----------------------
Side to side


Side to side at eye level
----------------------
Eye level to chin

The closer those measurements are to Phi (approximately 1.6180339887498948482), the closer to perfect is the face.

Part I: Let's try it!

To look at the photos, open the Java Applet Index page.

Note: It will open in a separate window. Arrange your browser windows so that the applet and the student page can be easily viewed.

Here are some things to notice on the applet:

  • display of x (in pixels)
  • display of y (in pixels)

To get an idea of how this will work, select Annie's Face.

Here are some questions to think about:

Looking at the x-coordinates
  1. Click in the center of her left eye. What value is x?
  2. Click in the center of her right eye. What value is x?
  3. What is the difference between the two values of x?
Looking at the y-coordinates
  1. Click on the top of her head. What value is y?
  2. Click on the bottom of her chin. What value is y?
  3. What is the difference between the two values of y?

Measurements:

Now we're ready to find some measurements. As you measure using the applet, record the x or y coordinate of the points. If the line segment is horizontal, pay attention to the x-coordinates. If the line segment is vertical, pay attention to the y-coordinates.

These are the measurements we got for the larger rectangle:

y coordinate at the top of head = 357
y coordinate at the bottom of the chin = 25
difference = 332

x coordinate on the left side = 261
x coordinate on the right side = 55
difference = 206

Are yours similar?

These are the measurements we got for the smaller rectangle:

y coordinate at eye level = 179
y coordinate at the bottom of the chin = 25
difference = 154

x coordinate on the left side = 261
x coordinate on the right side = 55
difference = 206

Are yours similar?

Using the data: Now we just need to calculate the ratios. Using the data we found, these are our ratios.

   332                          206
   ---  =  1.6116504    and     ---  =  1.3376623
   206                          154
 
How do your calculations compare?

Part II:

Now that you have the idea of how to find the two ratios, find the measurements for the other faces and decide which one is the most perfect.

Final Task:

Generalize the method that you used to find the lengths between points located on the face. Write the steps needed to calculate the two different ratios.

Extension:

Use your own photograph and see how perfect your face is!

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Selections from Ask Dr. Math Archives:

Appearances of the Golden Number
Why does the irrational number phi = (1 + sqrt(5))/2 appear in so many biological and non-biological applications?

FAQ: Golden Ratio, Fibonacci Sequence
Please tell me about the Golden Ratio (or Golden Mean), the Golden Rectangle, and the relation between the Fibonacci Sequence and the Golden Ratio.

Phi
What is phi?

phi vs. Phi - a Coincidence?
Ancient and modern architecture reflect the 'golden ratio' (1.618. length to width) and this number is remarkably close to phi (.618...) seen in nature for leaf dispersions, etc. Is this just a coincidence?

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