Reflecting: map onto

The last tutorial named a line so you could reuse it. This tutorial introduces map — the way you ask Beloch for the crease that carries one point or line onto another. It's the operation almost every fold is built from. (We're still just marking these creases; actually folding along them is the next tutorial.)

Folding a point onto a point

; midline.bel
paper square
mark --v = map .a onto .b

map .a onto .b is an expression: it builds the crease that, if you folded along it, would carry .a exactly onto .b. .a is the bottom-left corner and .b the bottom-right corner, so the crease that maps one onto the other is the vertical line straight down the middle of the sheet. We name it --v and mark it, same as before.

map onto is the single most-used building block in Beloch. Almost every crease you'll ever make is really an answer to the question "what fold carries this onto that?" — it's the fold itself, expressed as the two things it lines up.

Folding a line onto a line

map isn't limited to points — it also works on lines.

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3
4; bisect.bel
paper square
mark --v = map .a onto .b
mark map --v onto --ab toward .a

map --v onto --ab asks for the crease that carries the line --v onto the bottom edge --ab — in other words, the line that bisects the angle between them. But there are two lines that do that: one running toward the .a corner, and one running toward the mirror image of .a on the other side.¹ map alone can't tell which one you mean.

toward .a resolves the ambiguity by naming a point on the side you want. Whenever a map expression has more than one crease that satisfies it, toward <point> is how you pick.

What you learned

• map .p onto .q is an expression that builds the crease which carries point .p onto point .q by reflection.
• map also takes lines: map --l onto --m builds the crease that carries --l onto --m.
• Some map expressions have more than one valid crease. toward <point> disambiguates by choosing the one on that point's side.

Next, we'll look at the difference between drawing a crease you can see and actually folding the paper along it: mark vs. fold.

Footnotes

1. Two lines crossing always make two angle bisectors, perpendicular to each other — one for the angle on each side of the crossing. ↩