Skewed Ellipses

My diurnal age today is 26793 days and one of the properties of this number is that it's a member of OEIS A118886:

A118886

Numbers expressible as \(x^2 + xy + y^2 \), \(0 \leq x \leq y\), in 2 or more ways.

So in the case of 26793, this means that there are at least two values for \( (x,y) \) such that \(x^2 + xy + y^2 =26793\). It turns out that there are exactly two such values and they are (9, 159) and (93, 96). It's easy to forget that the equation  \(x^2 + xy + y^2 =26793\) is that of an ellipse that is skewed with respect to the \(x\) and \(y\) axes. In other words, it's not your more usual  \(x^2 + y^2 =a^2\) because the \(xy\) term skews it. 

Figure 1 shows the graph along with the two points that have positive integer values: A = (9, 159) and B = (93, 96).

Figure 1

These sorts of ellipses \(x^2 + xy + y^2 =a^2\) have axes of symmetry of \(y=x\) and \(y=-x\) and \( \pm a\) are the intercepts on the \(x\) and \(y\) axes. In the graph shown in Figure 1, the intercepts are \( \pm \sqrt{26793} \approx 163.7 \).

Of course, there are other points on the graph that have integer values but some of these are negative and so are not included in OEIS A118886, Without the condition that \( x \leq y), other points will arise as well. These full range of eight points is shown in Figure 2 and the graph with points added in Figure 3.

Figure 2

Figure 3

Now up to 27000, there are 2043 numbers that qualify for membership in  OEIS A118886 and that represents about 7.6%. However, there are only 69 numbers that are square numbers as well meaning that the intercepts of the \(x\) and \(y\) axes have integer values. These values are:

49, 169, 196, 361, 441, 676, 784, 961, 1225, 1369, 1444, 1521, 1764, 1849, 2401, 2704, 3136, 3249, 3721, 3844, 3969, 4225, 4489, 4900, 5329, 5476, 5776, 5929, 6084, 6241, 7056, 7396, 8281, 8649, 9025, 9409, 9604, 10609, 10816, 11025, 11881, 12321, 12544, 12996, 13689, 14161, 14884, 15376, 15876, 16129, 16641, 16900, 17689, 17956, 19321, 19600, 20449, 21316, 21609, 21904, 22801, 23104, 23716, 24025, 24336, 24649, 24964, 25921, 26569

Take 26569 as an example (see permalink). We have \(26569=159^2\) and the two \( (x,y) \) that satisfy  OEIS A118886 are (0,163) and (75, 112). The full range of ten points are:

(-163, 0), (-163, 163), (-112, -75), (-75, -112), (0, -163), (0, 163), (75, 112), (112, 75), (163, -163) (163, 0)

With these skewed ellipses, if the \(a^2\) in \(x^2 + xy + y^2 =a^2\) is a square number, then there are always another two integer-valued points, \( (a, -a) \) and \( (-a,a) \), compared to non-square numbers. 

Up to 27000, the maximum number of points that satisfy \( 0 \leq x \leq y \) is six. The numbers are 12103, 19747, 22477 and 23569. Using 12103 as an example, the points are (2, 109), (21, 98), (27, 94), (34, 89), (49, 77) and (61, 66). Even extending the range to 100,000, there are no numbers that yield seven points which is not surprising considering the symmetry of the graph. 

There are however, four numbers that yield eight points and these are 53599, 63973, 74347 and 84721. Using, 53599 as an example the eight points are (3, 230), (25, 218), (43, 207), (58, 197), (85, 177), (90, 173), (102, 163) and (122, 145).

Reuleaux Triangle

Today, having turned 26518 days old, I found an interesting property of this number that qualifies it for membership in OEIS A340644:

A340644

The number of vertices on a Reuleaux triangle formed by the straight line segments mutually connecting all vertices and all points that divide the sides into \(n\) equal parts.

 The initial members of the sequence are:

3, 19, 120, 442, 1332, 2863, 5871, 10171, 17358, 26518, 40590, 57757, 81735, 110209, 148158, 192184, 248772, 313105, 393429, 483283, 593490, 715528, 861660, 1022281, 1211811, 1418515, 1659108, 1919842, 2220204, 2543527, 2912751, 3308305, 3755922, 4233730, 4770150, 5340529, 5977071

26518 corresponds to the case where \(n=10\) and Figure 1 illustrates the situation:

Figure 1: source

The vertices referred to are best illustrated in the simplest non-trivial case of \(n=2\) where 19 vertices can be counted. See Figure 2.

Figure 2: source

The Reuleaux triangle has some very interesting properties but first a definition from Wikipedia:

A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its centre on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?"

Figures 3 and 4 illustrate what is mentioned in the definition:

Figure 3: Wikipedia

Figure 4: Wikipedia

The Reuleaux triangle has the smallest possible area of any curve of given constant width. This area is:$$ \frac{1}{2}(\pi - \sqrt{3}) \, s^2 \approx 0.77477 \, s^2 \text{ where } s \text{ is the constant width}$$At the other extreme, the curve of constant width that has the maximum possible area is a circular disk, which has area given by:$$ \frac{\pi \, s^2}{4} \approx 0.78540 \, s^2$$Wikipedia goes on to say that:

Any curve of constant width can form a rotor within a square, a shape that can perform a complete rotation while staying within the square and at all times touching all four sides of the square. However, the Reuleaux triangle is the rotor with the minimum possible area. As it rotates, its axis does not stay fixed at a single point, but instead follows a curve formed by the pieces of four ellipses. Because of its 120° angles, the rotating Reuleaux triangle cannot reach some points near the sharper angles at the square's vertices, but rather covers a shape with slightly rounded corners, also formed by elliptical arcs. Figure 5 illustrates this.

Figure 5

Interestingly, we also learn from the same source that:

Many guitar picks employ the Reuleaux triangle, as its shape combines a sharp point to provide strong articulation, with a wide tip to produce a warm timbre. Because all three points of the shape are usable, it is easier to orient and wears less quickly compared to a pick with a single tip.

Figure 6: source

To quote from the site:

APC™ Trident Pick

$35.00

The APC™ Trident is our take on an historic shape used by acoustic guitar players, electric players as well as mandolin players. This shape also has an interesting property, it is a curve with a consistent width also known as a Reuleaux triangle. The world’s first guitar pick that is true Reuleaux triangle. It can roll smoothly.

Another application of the shape, to quote again from Wikipedia, is:

The Reuleaux triangle has been used as the shape for the cross section of a fire hydrant valve nut. The constant width of this shape makes it difficult to open the fire hydrant using standard parallel-jawed wrenches; instead, a wrench with a special shape is needed. This property allows the fire hydrants to be opened by firefighters (who have the special wrench) but not by other people trying to use the hydrant as a source of water for other activities. Figure 7 illustrates this.

Figure 7

Clifford Pickover in his famous "The Math Book" has a section on the Reuleaux Triangle in which he writes:

Reuleaux Triangle 

Franz ReuIeaux (1829-1905)

The Reuleaux triangle (RT) is one example of a wide class of geometrical discoveries like the Mobius Strip that did not find many practical applications until relatively late in humankind's intellectual development. Not until around 1875, when the distinguished German mechanical engineer Franz Reuleaux discussed the famous curvy triangle, did the RT begin to find numerous uses. Although Reuleaux wasn't the first to draw and consider the shape formed from the intersection of three circles at the comers of an equilateral triangle, he was the first to demonstrate its constant-width properties and the first to use the triangle in numerous real-world mechanisms. The construction of the triangle is so simple that modem researchers have wondered why no one before Reuleaux had exploited its use. The shape is a close relative of a circle because of its constant width, meaning that the distance between two opposite points is always the same.

Various technology patents have focused on drill bits that cut square holes using the RT. At first, the notion of a drill that creates nearly square holes defies common sense. How can a revolving drill bit cut anything but a circular hole? But such drill bits exist For example, the illustration shown here is from the 1978 patent U.S. 4,074,778 for a "Square Hole Drill" and is based on the RT. The RT also appears in patents for other drill bits as well as for novel bottles, rollers, beverage cans, candles, rotatable shelves, gearboxes, rotary engines, and cabinets.

Many mathematicians have studied the Reuleaux triangle, so we know a lot about its properties. For example, its area is \( A = \frac{1}{2}(\pi - \sqrt{3}) \, r^2 \) and the area drilled by a RT drill bit covers 0.9877003907... of the area of an actual square. The small difference occurs because the Reuleaux drill bit produces a square with very slightly rounded corners. Figure 8 illustrates the patent design.

Figure 8

The Reuleaux Triangle also appears is architecture, especially Church architecture where it represents the Trinity. Figure 9 depicts a Reuleaux triangle shaped window of Sint-Salvatorskathedraal, Bruges.

Figure 9: source

Figure 10 shows the Trefoil knot, associated with the Borromean Rings. The Reuleaux Triangle is in the centre together with the three overlapping each being a vesica pisces.

Figure 10

This is just a small sampling of the properties and applications relating to the Reuleaux Triangle and much more could be written but I'll have to stop somewhere. 

2017: A New Year

As the new year begins, it seems appropriate to look at the mathematical character of the number that will identify it: 2017. As a start, this number is prime, in fact the 306th prime. It's nearest neighbours are 2011 and 2027. Thus there will not be another prime year for a decade. Working through the Online Encyclopaedia of Integer Sequences (OEIS), I was made aware initially that the number was linked to one circle and 23 ellipses, each with the major axis equal to the diameter of the circle and major/minor axes coincident with the x and y axes.

The equation all twenty four shapes have in common is \(ax^2+bxy+ cy^2 = 2017\).

In the case of the circle, \(a=1, b=0, c=1\) and so \(x^2 + y^2 = 2017\). The integer solutions are \(x=9\) and \(y=44\). This is the point A on the circle c in the diagram below.

For the ellipses, \(a=1, b=0\) and so \(x^2 + cy^2 = 2017\). The values of \(c\) for which there are integer solutions to \(x\) and \(y\) are 2, 3, 7, 14, 21, 24, 27, 31, 33, 42, 46, 56, 66, 81, 84, 87, 88, 93, 112, 232, 253, 462 and 1848. I've plotted the cases of \(c\)=2, 3, 7 and 14 in the diagram below. The associated points are (37, 18), (17, 24), (15, 16) and (1, 12).

 

For the equation \(ax^2+bxy+ cy^2 = 2017\), when \(a\) is not equal to 1 but \(b\) is still 0, the following \((a, b, c)\) values give integer solutions to \(x\) and \(y\): (4, 0, 9); (2, 0, 41); (5, 0, 17); (2, 0, 65); (2, 0, 95); (8, 0, 65). All these ellipses lie inside the circle \(x^2+y^2=2107\) and have been graphed below (with part of the surrounding circle visible):

If \(xy\) terms are allowed, then there is another whole series of ellipses of the form \(ax^2+bxy+cy^2 = 2017\) where the following values of (a, b, c) yield integer solutions for x and y: (1, 1, 1); (9, 6, 1849); (1, 26, 1); (1, 8, -8); (1, 10, 1); (2, 1, 3); (1, 1, 8); (1, 1, 22); (4, 1, 4); (4, -1, 4); (4, -3, 5); (2, -1, 10); (2, 1, 12); (2, -1, 12); (1, 20, 1); (1, 31, 1); (8, 8, 97); (37, 4, 37); (28, 12, 57); (57, 18, 193). A few of these I've plotted below (with the circle included for comparison):

2017 can also be written as the sum of three (not distinct) cubes, namely \(7^3+7^3+11^3\). Thus it seems that 2107 is unusual in that it can be linked in 2-D space to a large numbers of ellipses. Of course, ellipses are the orbits followed by celestial bodies within the solar system trapped by the gravitational forces of larger bodies such as the Sun and planets.

Consider one of the ellipses above, say \(4x^2+9y^2=2017\) and a general point \((x, y)\) situated on it. The only integer values of \(x\) and \(y\) that satisfy this equation are (±22, ±3) as shown below: