Besides that the kites to induce grid jumps, they provide more surprises. Each kite has a complementary partner kite where both kite angles adds to 90°. Consequently its possible that both the kite and its partner can occur in the same tiling of a form. I found the idea and a lot of other details in [MathPuzzle] Kites&Bricks .
In the following we will see, that there are infinitely many kite pairs. They are tightly connected with [M1] Pythagorean Triples in a very interesting way. Furthermore some generalisations are possible.
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The two kites of type 2-1 and of type 3-1 complement one another to the angle of 90°, because atan(1/2) + atan(1/3) = π/4 is valid. At the same time both angles are equal to the angles of the Pythagorean triangle 3–4–5, because 2*atan(1/2) = atan(4/3) and 2*atan(1/3) = atan(3/4) are valid. These connections are not at all accidental, but are a general property of all kite pairs. The picture on the right shows a tiling of a 7*7 square with kites of type 2-1 and 3-1 and 2*1 rectangles. The tiles fit perfectly even if its not really obvious. |
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The two kites with the legs 2-3 and 5-1 complement one another for a right angle too. With some calculation with elementary trigonometric formulas and usage of the well known tangent addition theorem we get:
Its obvious that all kites double the angle of the defining triangle by construction, so we get:
This means, that the kite angle α is always equal to the angle of a Pythagorean triangle, which is defined by a well known formula (picture left) from the parameter n and m. This is valid for all relatively prime natural numbers n and m, where is m < n and only one of them odd. This way the parameter n and m get a smart geometric interpretation. This says immediately, that there is a exactly one kite pair for each of the infinitely many Pythagorean triangles. The only point left is, whether or not all pairs meet on a grid point of the diagonal line. This follows by applying the addition theorem of the inverse tangent:
That the angle α and β of the kite pair in the left picture add up to 90°, is made by construction already, but the above formula allows the interpretation, that the legs of the second kite can be derived from n - m and n + m and therefore are whole numbers.
Because of n - m and n + m are odd numbers, they are not suitable as parameters for the Pythagorean formula. That means the relationship between the two kites of a pair is not symmetric. When we start with odd m and n and go the opposite direction, we get with n - m and n + m numbers with unequal parity. So by changing roles we get valid parameters for the Pythagorean formula. That means that all kite pairs correspond one to one to a Pythagorean triple.
Each pair has therefore a main kite derived from a Pythagorean triple and a complementary kite.
This kite develops from the parameter m = 2 and n = 5. We see that the Pythagorean triangles defining the angle can grow pretty large. I briefly mention other interesting properties of the Pythagorean triangles like incircles with whole number radius, touching three sides at grid points.
Most people know about Pythagorean triangle that even in the old days you could construct right angles. That is true without doubt, but for me the real magic comes from another fact. You can find points where the diagonal connection line length is exactly an integer value! That this line can be completed to a right angel triangle is simple and comes out of the nature of the square grid.
When we apply the same idea on a regular triangle grid and connect two grid points diagonally, it's even more surprising that we can get an integer length line. The height of the grid elements is √3/2 and not at all a rational number, but such connections are indeed possible. Once we have such a diagonal integer line, we can complete a 120° triangle naturally. The smallest of such construction is a 3–5–7 triangle. I found the corresponding formulas in an article (German) [K2] Hoehn/Walser, concerned with other properties of 120° triangle.
The following calculations show, that every Pythagorean 120° triple corresponds to a kite pair in the triangle grid. The properties resemble the kite pairs of the square grid.
All Pythagorean 120° triples can be generated similar to the 90° triples by a formula with two basic parameters. The above example shows the result of n = 2 and m = 1.
The triangle kite complement one another to 60° with its partner, where the legs of the partner kite are n - m and n + 2*m in length. We then can calculate the half angle of the kite using elementary trigonometric formulas.
After a lengthy calculation and applying the arctan addition theorem we find that the constructed kites have angles that complement to π/3 = 60°.
Very similar to the quadratic case there is a close connection of the kite pairs to Pythagorean 120° triples. The angle of the primary kite is equal to the angle of the Pythagorean 120° triangle, which is produced from the parameter n and m by the formula above. The values n and m have the same way a clear geometric meaning as length of the kite legs. This is verified by the following calculation:
Because the angle of the kite and the angle of the corresponding Pythagorean 120° triangle have equal tan values and also reside in the same quadrant, we know that the angles are equal.
This picture shows an additional example of a triangle based kite pair. It is derived from the parameter n = 3 and m = 2 following a = 5, b = 16 and c = 19 .
