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Squaring

Squaring draws a squared rectangle, which is a rectangle tiled by squares, next to its Smith diagram. In that circuit every horizontal segment is a node and every square is a unit resistor whose current equals its side.

Use this type​

Use Squaring to explore squared rectangles and squared squares, or to check whether a dissection is simple and perfect. Sketch one with unknown sides and let the solver close it. Or build one backwards from a planar network by choosing or searching the battery edge.

Source format​

The source format is Squaring text. Enter the source in the diagram.zip editor.

Syntax essentials​

  • Optionally begin with .view rectangle, .view circuit, .view overlay, or .view both; the default is both.
  • Add .labels none to hide side lengths, voltages, and currents.
  • Add title followed by text to name the diagram.
  • For a known dissection, write rectangle <width> x <height> and then the square sides in Bouwkamp order; parentheses and commas are ignored.
  • Bouwkamp order places each square at the leftmost point of the highest unfilled segment. Squares that share a top edge are listed left to right.
  • Write rectangle ? x ? to infer the size; the first group is then the top row.
  • A ? side is a square as wide as the gap it lands in. Squares that do not fit stop placement, and the rest of the rectangle is hatched.
  • For a sketch, name sides with letters or expressions such as a, 2a, or a+b. Every later group needs an under clause such as (g h) under f c, listing the squares it rests on from left to right.
  • The sketch solver scales the unique solution to whole numbers, or reports free lengths, contradictions, or a fractional side.
  • For a network, write battery <positive> <negative> and then wires as wire a b, chains such as a - b - c, face a b c d polygons, or polyhedron cube.
  • Write battery any to try every wire as the battery; the best result is drawn and every candidate is listed.
  • The battery replaces one edge of the graph, so do not list that edge as a wire. A wire parallel to the battery becomes a full-height square.
  • Every wire has resistance one; the renderer solves Kirchhoff and Ohm exactly, scales the voltages so all sides are whole numbers, and arranges the squares.
  • The network form needs a planar network in which the battery closes a face; polyhedral (3-connected planar) networks give a simple squaring.
  • Symmetric solids often fail for every battery because mirror-image nodes settle at the same voltage; asymmetric polyhedra work best.
  • The caption reports the order, size, whether the squaring is simple (no smaller rectangle of squares) and perfect (no repeated side), and the battery.
  • Compound blocks are outlined; node color runs from blue at the negative pole to red at the positive pole.
  • Start comments with #.

Example​

.view both
title Order 9 simple perfect squared rectangle

rectangle 33 x 32
squares (18 15) (7 8) (14 4) (10 1) (9)
Rendered exampleDrag to pan. Use the wheel or controls to zoom.
100%
This view uses the live diagram.zip renderer.

Origin​

Squaring originates with Brooks, Smith, Stone, and Tutte at The dissection of rectangles into squares (Duke Mathematical Journal, 1940). Diagram.zip implements a text notation and SVG renderer for the squared-rectangle and Smith-diagram correspondence that the four Trinity students discovered.

Limitations​

  • Networks are arranged by search, so very large or non-planar networks are rejected instead of drawn.
  • A wire whose two nodes settle at the same voltage carries no current and is rejected; choose another battery or use battery any.
  • battery any is limited to networks with at most 60 wires.
  • Node positions in the Smith diagram follow the midpoints of the horizontal segments rather than a spring layout.

Upstream reference​

Squaring documentation