Parallel lines and regular hexagons

Once you can copy an angle, you can build a whole world: parallel lines, regular hexagons, and the stars and flowers tiled across temple ceilings.

Idea

Parallel lines via copied angles. Recall from Chapter $5$: if a transversal cuts two lines making equal corresponding angles, the two lines are parallel. We turn this into a construction.

Given a line $m$ and a point $P$ not on $m$, we want to draw a line through $P$ parallel to $m$.

Construction.

1. Draw any transversal line $\ell$ through $P$ that crosses $m$ at some point $A$.
2. At $A$, the lines $\ell$ and $m$ make some angle $\alpha$.
3. At $P$, copy this same angle $\alpha$ on the same side of $\ell$, using the angle-copying construction.
4. The new ray extends to a line $n$ through $P$.
5. Since the corresponding angles between $\ell$ and the two lines $m, n$ are equal, $m \parallel n$.

Regular hexagons. A regular hexagon has six equal sides and six equal interior angles. Each interior angle is $120°$, and the centre angle for each of its six equilateral wedges is $60°$.

The neat fact: a regular hexagon is exactly six equilateral triangles glued together at a shared centre. So once you can construct an equilateral triangle (and you can, via the $60°$ construction), you can build the whole hexagon.

Construction (radius $r$).

1. Draw a circle of radius $r$ centred at $O$.
2. Mark any point $A$ on the circle.
3. With centre $A$ and radius $r$, cut the circle at $B$. With centre $B$ and radius $r$, cut at $C$. Continue around: $D, E, F$. The sixth arc returns to $A$.
4. Join $ABCDEF$ in order. This is a regular hexagon with side $r$.

Why exactly $6$? Each chord $AB, BC, \ldots$ equals the radius $r$, and each of the six central triangles $OAB, OBC, \ldots$ is equilateral. Six $60°$ angles around $O$ add up to $360°$ , a complete turn , so the hexagon closes perfectly.

Beautiful designs from this. The seed-of-life flower, six-petal mandalas, the eight-pointed star (rotate two squares), the Sikri jali patterns , all start from circles of the same radius arranged around a centre. Try one!

Worked examples

Example 1. Draw a line $m$. Mark a point $P$ above it. Construct a line through $P$ parallel to $m$.

Draw a slanted transversal through $P$ meeting $m$ at $A$. Copy the angle made at $A$ (between $m$ and the transversal) to $P$, on the same side. Extend the new ray , that is the parallel.

Example 2. Construct a regular hexagon of side $4$ cm.

Draw a circle of radius $4$ cm. Pick a point $A$ on it. Stepping the same $4$ cm radius around the circle, mark $B, C, D, E, F$. Join in order. The hexagon has all sides $4$ cm and all angles $120°$.

Example 3. Why are the central triangles of a regular hexagon equilateral?

Each central triangle has two sides equal to the radius (both joining centre to a vertex), and the third side is a chord that also equals the radius (by construction). So all three sides are equal , equilateral.

Example 4. Construct a six-petalled flower.

Draw a circle of radius $r$. With the same radius, draw a circle centred at each of the six vertices $A, B, C, D, E, F$ of the inscribed hexagon. Where each neighbour-pair of circles intersects, a "petal" forms. You will see exactly six petals around the centre.

Try it yourself

1. Draw a line and a point above it. Construct the parallel through that point.
2. Construct two parallel lines $4$ cm apart.
3. Construct a regular hexagon of side $5$ cm.
4. Construct a regular hexagon and join all three pairs of opposite vertices. What shape do you get inside?
5. Inscribe an equilateral triangle in a circle of radius $4$ cm. (Hint: connect every alternate vertex of the inscribed hexagon.)
6. The interior angle of a regular hexagon is $120°$. Show this using the fact that the sum of interior angles of a polygon with $n$ sides is $(n-2)\times 180°$.
7. Combine a regular hexagon and six outward equilateral triangles to make a six-pointed star.
8. Why does the $6$-step procedure around a circle close back to the starting point exactly?

Activity

Take three $\rupee 5$ coins (or three identical bottle caps). Place them so each touches the other two. Now place three more on the outside so each touches two of the inner ones. You have just physically constructed a hexagonal arrangement , the same one bees use for honeycomb. Why is hexagonal packing so efficient?