A Rope Can Do What a Compass Does Constructions and Tilings

Constructing a perpendicular bisector, the almond-eye-shaped construction familiar from Class 6, is really two chained congruent-triangle proofs in disguise: drawing equal-radius arcs from both X and Y above and below the segment creates two crossing points, and since all four arc-radii used from X and Y are equal, the triangle formed by X, Y, and either crossing point has two equal sides (matching the equal radii) plus the shared base XY, making the two triangles either side of XY congruent by SSS -- which forces the angles at each crossing point to split evenly, and a second congruence (SAS this time) then proves the line joining the two crossing points meets XY at a perfect right angle, exactly at its midpoint.

The Sulba-Sutras, ancient Indian geometric texts from the Vedic period used for constructing fire altars, contain the earliest known versions of these exact same construction methods -- but performed using a rope rather than a compass. The Katyayana-Sulbasutra (verse 1.2) describes fastening the two loops of a doubled rope to pegs at points X and Y, then pulling the rope's midpoint taut, first above the line and marking point A, then below it and marking point B -- the line AB produced this way is exactly the perpendicular bisector of XY, since the taut rope guarantees equal distance from both pegs at every position, just as a compass's fixed radius does.

Bisecting an angle uses the same underlying idea: one arc from the vertex marks two equidistant points on the two arms, and then two more equal-radius arcs from those two points cross at a spot that -- by an SSS congruence between the two halves -- is guaranteed to sit exactly on the angle's bisector. Copying an angle exactly, without ever measuring it in degrees, uses a closely related trick: an arc from the original angle's vertex marks two points on its arms, the same-radius arc is then drawn at the new location, and finally the exact arc-distance between the two original marked points is transferred (via compass) onto the new arc, fixing the second ray at precisely the same angle as the original -- proven identical by SSS congruence between the two small triangles this creates.

Copying angles this way makes it possible to construct a line parallel to a given line through any point (by reproducing a matching corresponding angle), which in turn builds real architectural forms: the trefoil arch and pointed arch seen at Delhi's Red Fort (Diwan-i-Aam) are both constructed from equal-length, equal-angle support lines using exactly these techniques. Building a regular hexagon works similarly, using six equilateral triangles arranged around a shared centre point -- each equilateral triangle contributing its own 60 degree angle, and six of them meeting exactly at the centre point because six times 60 degrees comes out to exactly 360 degrees, filling the space around that point perfectly with no gap and no overlap.

Whether a grid can be completely covered by simple two-square domino tiles depends entirely on counting: a 4-by-7 grid has 28 squares (an even number, tileable), but a 5-by-7 grid has 35 squares (an odd number), and since every domino tile covers exactly 2 squares, an odd total can never be exactly covered -- no tiling is even possible before a single tile is placed. A more powerful version of this same idea colours the grid like a checkerboard: since every single domino, wherever it's placed, always covers exactly one black and one white square, any region that CAN be fully tiled must always contain an equal number of black and white squares -- so a region with, say, 8 white squares and only 6 black squares can be proven impossible to tile, without ever needing to try a single arrangement.

Squares, equilateral triangles, and regular hexagons can each tile an entire flat plane with no gaps or overlaps, repeating forever in every direction -- the same principle bees and wasps use naturally when building their hexagonal hives. The Dutch artist M.C. Escher turned this mathematical idea into striking interlocking-animal artwork, and the search for new ways to tile a plane is still active mathematics today: one particular tiling pattern, using a single repeating tile shape rather than needing two or more different shapes together, was only discovered as recently as 2023.