Naming Points & Lines

The previous tutorial used names Beloch gave you for free: .a .b .c .d for the square's corners, --ab --bc --cd --da for its edges. This tutorial shows how to give names to things you compute, so you can build on them without re-deriving the same line twice.

Naming a line

Here is the diagonal from the last tutorial, but this time we name it.

; diagonal.bel
paper square
mark --ac = through .a .c

--ac = through .a .c is a binding: it evaluates the expression on the right (through .a .c, the line through .a and .c) and attaches the name --ac to the result. .a and --ab were names the paper handed you; --ac is a name you chose. mark still does the same job as before — it records the line as a visible crease — it just now also has a name attached to it.

Reusing a name

The point of naming a line is that you can refer back to it later.

1
2
3
4; perp.bel
paper square
mark --ac = through .a .c
mark perp --ac through .b

perp --ac through .b is another expression: it builds the line perpendicular to --ac, passing through .b. It does not re-derive the diagonal — it looks up --ac by name and uses it directly. Without the name, this line would have to be written out as perp (through .a .c) through .b every time you needed it, and if the diagonal itself ever changed, you would have to find and update every place that spelled it out again.¹

What you learned

• --name = expression binds a name to a computed line; .name works the same way for points.
• Names you bind are no different from names the paper gives you — --ac and --ab both work anywhere a line is expected.
• Naming lets you build new expressions, like perp <line> through <point>, out of earlier results instead of re-deriving them.

Next, we'll use a named line as an axis to fold the paper onto itself: Reflecting: map onto.

Footnotes

1. This is the same reason you don't copy-paste a formula across a spreadsheet instead of naming a cell — one definition, referenced wherever it's needed. ↩