1. Applied Math in Haircutting
The Celestial System Approach

2. Hair Cutting
In the artist world, hair cutting is considered an art form, like a sculptor carving out the desired shape.
At a closer look, one will find that basic geometry and trigonometry play a vital role in the fundamentals of hair cutting

3. Art and Science! Art in hair cutting: A 3-D form to sculpt
Includes: hair lengths, textures, growth patterns, density, and elasticity which all determine the outcome of a hair cut.
Cosmetologists must learn to combine the art and creativity involved in haircutting with the fundamentals of geometrical angles, shapes in trigonometry, and the principles of applied math.

4. Celestial Axis in Haircutting
Celestial Axis - an imaginary sphere used as a background for locating positions in space.
Use in hair cutting to Identify straight and curved lines formed around the shape of the head

5. Celestial Axis as a Guide in Hair Cutting
The celestial axis assists in…
identifying parts, sections, angles, degrees, and the direction of movement.
The axis provides a guide to create various hair cutting forms.
Show students, using the mannequin, how to use the celestial axis

6. Celestial Axis on Head Form
For reference points in hair cutting, imagine a grid on the scalp.
Using the imaginary grid as a guide, lines, angles and degrees can be identified on the head.
Use the tools and mannequin to demonstrate

7. Geometry in Haircutting
Horizontal lines run parallel, (east and west), creating the illusion of added weight.
Example: One Length Haircut
Vertical lines run perpendicular (north and south), creating the illusion of removing weight.
Example: Layered Haircut.
Teacher Note: Show the differences between horizontal and vertical lines on the mannequin’s scalp

8. Geometry-Creating Diagonal Lines in Haircutting
Diagonal Lines run between horizontal and vertical lines
create the illusion of less weight in a haircut.
Diagonal Lines in haircutting create a angle, depending on finger position.
Beveling - a technique used in diagonal lines to create a stack appearance in hair.

9. Trigonometry in Haircutting
Trigonometry - the study of curved space, such as a head form.
Angles are used in haircutting as reference points of how to envision the hair before cutting.
Examples: 0°, 45°, 90°,135°,180°

10. Concave and Convex Lines in Haircutting
Concave Lines
Used to create a curve like an arch.
Most often used in the back of the head, around ears, and bang area, (also called fringe)
Convex Lines
used to create a “U” or “V” shape in haircutting.
The center of the convex line creates the illusion of weight.
The area of weight in any haircut will be the focal point.
One of the goals in haircutting, is to maximize a clients good features and minimize any poor features.
Convex lines can minimize broad shoulders.

11. Angles Used in Haircutting
Creates...
Illusions
Examples
Zero
Horizontal lines in haircutting
1. Weight or fullness
2. Rectangle or oval form
One-Length Cuts
45°
A stack or wedge
Weight lines should reflect positive facial features
Triangle Forms
90°
Hair that is the same lengths in that area, called a uniform cut
1. More width
2. Make the face appear more round
Circular Forms
180°
Layered hair, start below the crown and progress to end length
Fullness and body
Oval Forms

12. Reference Points Used in Haircutting
Parietal Ridge- Widest area of the head.
Area used to measure for wig size, custom hats, and baby developmental growth.
Occipital Bone- Bone that protrudes at the base of the skull.
Apex- Highest point on the head.
Found by placing a comb flat on the top of the head. The highest point, is where the comb rest on the head, laying horizontally.
Four Corners- Shows where the head begins to curve on each side of the head, from flat to round.
Used mostly for determining reference points for bangs.

13. Reference Points of the Head
Crown- Area between the apex and occipital bone. Different hair growth patterns exist in this area, requiring caution when cutting hair.
Top- Area between the top of the forehead and the crown.
Sides- Area from the back of the ear towards the face and is below the parietal ridge.
Nape- Area below the occipital bone to neckline.
Point out all of the reference points on the head

14. Working Areas of the Head
Sections- Hair is parted in working areas for ease and better results in haircutting.
Subsections- Sections divided into smaller areas.
Parting- A line, horizontal, vertical or diagonal used to separate the subsection.
Elevation/Projection- The angle or degree the hair is held, in relation to the head.
Tension- Amount of pressure used when holding hair between fingers while cutting hair.
Teacher note: show students the different working areas of the head using the mannequin

15. Haircutting Lines
Cutting line- is the angle the fingers are held when cutting hair. Also called finger position.
Guideline- is the first cut taken to use as a reference for the rest of the hair. Also called the guide. Determines length that will be cut.

16. Two Types of Guidelines
Traveling Guideline- Taking a small piece of the previous hair cut to use as a guide for the next area to be cut. Also called movable guide.
Stationary Guideline- Sections are combed to one area to be cut. The guide does not move.
Teacher note: demonstrate both times of guidelines on the mannequin for the class

17. Distribution
Distribution - the direction the hair is combed from the parting.
Natural Distribution- the direction the hair naturally falls from the curve of the head.
Perpendicular Distribution- is achieved by combing the hair straight out from the head, using a 90° angle.
Shifted Distribution- Used for length increases and blending from one area to another.
Directional Distribution- The hair is combed straight up or out from the curve of the head. Length increase will be the result cut.

18. Head Position in Haircutting
Upright Head Position- Used for hair length
above the shoulders.
Forward Head Position- Used when a beveled haircut is desired. Recommended when cutting hair below the shoulders in the back area.
Titled Head Position- Used when working on the sides of the head for comfort and ease for the cosmetologists to better see the working area.
Use the mannequin to show the different head positions to the class

19. Partings Partings – Lines that subdivide the hair
so the hair is easier to control.
Partings are generally parallel to the guideline.
Partings can be horizontal, vertical, diagonal forward, diagonal back, concave, convex, curved or zigzag.

20. Finger Position Finger position is in relation to the parting.
The direction the hair is held.Two types are used:
Parallel Finger Position - Fingers are positioned same direction as parting.
Non-Parallel Finger Position- Fingers are positioned unequally to the part. The position used will determine the line cut.

21. Crosschecking Haircuts
Crosschecking is a term used to make sure the hair is cut correctly.
Check a haircut for balance and accuracy, use the opposite parting implemented in the haircut.
Example: If the haircut was performed with horizontal partings, check for accuracy using large sections and vertical partings.
Demonstrate crosschecking to the class using the mannequin

22. Work Cited Milady’s Standard Textbook, copyright 2004
Salon Fundamentals Textbook, copyright 2000