Chapter 14 — Constructions and Tilings

Chapter 14, Constructions and Tilings, covers two topics. The first is geometric constructions using only an unmarked ruler and compass — including perpendicular bisectors, angle bisectors, 60°/90°/45° angles, copied angles, parallel lines, and regular hexagons. The second is tiling — covering a region with shapes without gaps or overlaps — including proofs of when grids can or cannot be tiled.

Fix the same radius on a compass. From one end X of the line segment, draw arcs above and below the segment. Without changing the radius, repeat from the other end Y. Label the two arc-intersection points A (above) and B (below). The line AB is the perpendicular bisector. It passes through the midpoint of XY at a 90° angle.

Because AX = AY = BX = BY (equal radii). Any point equidistant from both endpoints of a segment lies on the perpendicular bisector. The chapter justifies this through congruence: triangles ∆ABX ≅ ∆ABY (SSS), which gives ∠XAO = ∠YAO, and then ∆AOX ≅ ∆AOY (SAS), so OX = OY and ∠AOX = ∠AOY = 90°.

Mark two points X and Y equidistant from the given point O on the line, so O is the midpoint of XY. Then draw one pair of intersecting arcs (on one side) from X and Y. Joining O to that intersection point gives a 90° angle at O, because the construction effectively draws the perpendicular bisector of XY through O.

Given angle ∠XOY, use a compass to mark A on OY and B on OX such that OA = OB. Then, keeping the same radius, draw arcs from A and B; let C be their intersection. The ray OC bisects angle ∠XOY. This works because ∆OBC ≅ ∆OAC by the SSS congruence condition, so ∠BOC = ∠AOC.

Draw an arc from the vertex A of the original angle, meeting the arms at B and C. At the new vertex X, draw an arc of the same radius meeting one arm at Z. Measure length BC with the compass and transfer it to get Y on the arc so that YZ = BC. By SSS congruence, ∆ABC ≅ ∆XYZ, so ∠A = ∠X.

Draw a transversal l through any point A on the given line m and through the new point B where the parallel line will pass. Copy the angle that l makes with m at point A, to point B on l. The line drawn through B at this copied angle is parallel to m, because equal corresponding angles imply parallel lines.

To get 60°, draw a line segment AB of any length, then with the same radius draw an arc from A and another from B; their intersection C gives ∆ABC equilateral, so ∠CAB = 60°. For a regular hexagon, use this 60° construction repeatedly: six congruent equilateral triangles arranged around a centre produce a regular hexagon because six 60° angles sum to exactly 360°, leaving no gaps.

First construct a 90° angle using the perpendicular bisector method, then bisect the 90° angle using the angle bisection steps. Bisecting 90° gives 45°.

Tiling means covering a region using copies of one or more shapes without any gaps or overlaps. For example, a rectangular grid can be tiled with 2 × 1 tiles. The chapter also discusses tiling the entire plane with shapes such as squares, equilateral triangles, and regular hexagons.

No. A 5 × 7 grid has 35 unit squares. Each 2 × 1 tile covers exactly 2 unit squares. Since 35 is odd, it cannot be divided evenly into groups of 2, so the grid cannot be tiled with 2 × 1 tiles.

Colour the grid like a chessboard so that every black square has only white neighbours and vice versa. Each 2 × 1 tile always covers exactly one black and one white square. Therefore, any region that can be tiled must have an equal number of black and white squares. If a region has an unequal count (as in the specific 5 × 3 grid with a removed square shown in the chapter, which has 8 white and 6 black squares), it cannot be tiled.

The chapter confirms that squares, equilateral triangles, and regular hexagons can each tile the entire plane on their own. The plane can also be tiled using combinations of shapes or non-regular polygons.

The Śulba-Sūtras are ancient Vedic geometric texts used for constructing fire altars for rituals; they are part of one of the six Vedāṅgas. They contain methods for constructing perpendiculars and perpendicular bisectors that match the ruler-and-compass methods in this chapter, but use a rope instead of a compass. The Kātyāyana-Śulbasūtra (1.2) describes a specific rope-based construction of the perpendicular bisector.

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