third method to tile the plane using a 4-gon – Discussion Forum

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Piyush365

Asked: October 5, 2026In: Class 9 Maths

Let us see a third method to tile the plane using a 4-gon. Focus on only two coloured copies of SOME sharing a vertex. What do you see? It appears that each 4-gon is just a shifted copy of the other. Let us see exactly how. Draw two copies of SOME as shown in Fig. 12.42 so that the diagonals EO and E’O’ are collinear with O = E’. Now place a cutout of one copy on top of SOME with diagonal EO drawn on it. Slide this cutout so that segment EO moves along EO’ until EO matches E’O’. Verify that your cutout exactly matches S’O’M’E’. (i) Prove the exact match of SOME with S’O’M’E’ using four parallelograms. (ii) Follow the described procedure along each diagonal of the starting coloured 4-gon in both directions to get 4 new 4-gons. Using these 4 copies, repeat the procedure 4 more times to place 4 new copies, resulting in a 3 by 3 grid of 9 copies exactly like the 9 coloured copies in Fig. 12.42. These 9 copies meet each other only at vertices and once again the blank spaces among them also form 4-gons congruent to SOME!

(i) The displacement vector mapping E to O (E’) defines four parallelograms: E E’ S’ S, E E’ O’ O and E E’ M’ M. Translations preserve edge lengths and angles, guaranteeing an exact congruent match. (ii) Translating along both ...