Geometrical Aspects of Folding, Origami
What geometry terms or examples did you notice while folding?
point: “a specific location” (no length or width, dimension zero)
line: “an unending length”, a breadthless length (Euclid), a widthless length
After Euclid’s work had been studied and analyzed, mathematicians decided it is impossible to define all terms without resorting to infinite regress. Point and line are considered to be officially not defined, just undefined words used without definition.
The basic building blocks. You need to start somewhere.
angles: right (90 degrees), acute (less than 90 degrees), obtuse (90 to 180 degrees), 45 degrees
parallel lines have equal distance between them, will never intersect
perpendicular lines form equal angles (90 degree right angles)
square: 4 equal sides and 4 right angles)
rectangle: opposite sides are equal and 4 right angles – note all 4 sides do not need to be equal length
parallelogram: opposite sides equal length and opposite angles are equal – note that all 4 angles do not need to be equal. A rectangle that gets pushed over to the side is a parallelogram.
Every square is a rectangle. True or False: If it is a square, then it is a rectangle. (True)
Not every rectangle is a square. True or False: If it is a rectangle, then it is a square. (False)
rhombus: 4 equal sides and opposite angles are equal – note that all 4 angles do not need to be equal. If you take a square and push it over, then you have a rhombus.
quadrilateral: any 4 sided polygon
triangle: any 3 sided polygon
equilateral triangle: 3 sides of equal length
isosceles triangle: 2 sides of equal length
scalene triangle: no sides of equal length(over)
Motions:
Flip or Reflect (pick up off table and turn over)
Rotate (keep flat on the table and turn like the hands of a clock
Convex: Fold edges “up”. The fold is away from you, and edges are toward you. Author Paul Jackson calls this a valley fold.)
Concave: Fold edges “back”. The fold comes toward you, and you move the edges of the paper away from you. Author Paul Jackson calls this a mountain fold.
Line of Symmetry: both parts look identical on either side of this line
How to make a square piece of paper by folding:
Method 1: Fold one corner down (keep edges even) until you have two isosceles 45-45-90 triangles on top of each other sharing the diagonal hypotenuse. Cut or nail crease and gently tear off the extra strip.
or
Method 2: Use a second piece of paper to mark the short side along the length of the long side. Crease. Cut or remove extra strip.