VectorPaint Euclidean square tiling: split eighty-four by thirty into seven integer squares

English translation of the Chinese original, translated on 2026-11-14. Original publication dates, attribution, code and version notes are retained.

Tile an 84×30 rectangle by repeatedly taking the largest square fitting the current remainder, matching quotients and remainders to Euclid’s algorithm. Check coverage and greatest common measure separately. VectorPaint organises static pieces without implying automatic tiling or greatest-common-divisor calculation.

Declare dimensions and one unit scale

Declare an 84-wide, 30-high rectangle with lower-left origin. On an 1800×1200 canvas, scale 15 gives displayed width 1260 and height 450, leaving dimension space. Separate outer frame, piece boundaries, identifiers and arithmetic. Units are invented lengths rather than physical millimetres. Euclid’s Book VII common-measure algorithm supplies the repeated-measure context. Integer rectangles and squares explain it here without implying a cutting plan or verified manufacturing dimensions.

Take two size-thirty squares first

84÷30 gives quotient 2 and remainder 24. Place square A at x=0..30 and B at x=30..60, both y=0..30. The right remainder spans x=60..84 and y=0..30: width 24 and height 30. Do not swap these dimensions while leaving the strip on the right. Draw the shared boundary once. Distinct light fills may identify A/B, but the declared scale and dimensions, not apparent colour widths, define the pieces.

Take a size-twenty-four square from the remainder

In the 24-wide, 30-high remainder, 30÷24 gives quotient 1 and remainder 6. Place square C at x=60..84,y=0..24. The remaining upper strip spans x=60..84,y=24..30, width 24 and height 6. The remainder changes from vertical strip to horizontal band, while algorithm inputs change from 84/30 to 30/24. Its horizontal boundary belongs only in the right remainder; extending it across A/B would silently split them and change the seven-piece count.

Divide the height-six strip into four squares

24÷6 gives quotient 4, remainder 0. Divide the band into D,E,F,G with x ranges 60..66,66..72,72..78,78..84 and y=24..30. Each is 6×6; four widths total 24 and close the remainder. Zero means the division is exact, not a reason to draw a zero-width square or seek another nonzero strip. Label each size once if four-edge repetition crowds the diagram; a clear dimension ledger is easier to audit.

Match quotients two, one, four to seven pieces

Write 84=2×30+24, 30=1×24+6 and 24=4×6+0. Quotients 2,1,4 mean two size-30, one size-24 and four size-6 squares, summing to seven. They do not mean seven size-six squares, and multiplying quotients does not give eight pieces. Last nonzero remainder 6 is gcd(84,30): 84/6=14 and 30/6=5. Those fourteen and five are rectangle dimensions in units of six, not this greedy tiling’s piece counts.

Audit coverage using area and boundary ledgers

Rectangle area is 84×30=2520. Piece areas 2×900+576+4×36 also total 2520. Equal area alone needs coordinate coverage, since overlaps and gaps could cancel. The seven square perimeters total 432, while the outer rectangle perimeter is 228. Difference 204 counts internal shared boundaries twice, giving internal independent length 102. Do not label 432 as outer perimeter. Trace the entire exterior, then verify the six internal separator segments and their shared identities.

Troubleshoot wrong remainders and stretched pieces

A first remainder of 14 would cover only x=60..74, exposing a missing width 10. If the upper strip is labelled height 24, check confusion between remainder 6 and current divisor 24. Resize the whole diagram uniformly rather than stretching a piece to fit its label, which would turn a square into a rectangle. Stroke widths are not cutting loss: contacts are zero-thickness mathematical boundaries. Real cutting requires separate process data absent here; static coverage does not establish usable physical dimensions.

Reopen and audit three equations and seven pieces

Save, leave and reopen. Hide dimension answers, recover A..G from coordinate boundaries, then write quotients 2,1,4 and remainders 24,6,0. Check common measure 6, area 2520 and exterior perimeter 228, excluding internal boundaries. Inspect short separators in the upper band and mathematical-unit scope in final output, leaving unexamined formats unverified. Readers should rebuild this arrangement and distinguish common measure, piece count and units. No claim is made that the greedy arrangement proves a minimum square count among all tilings.