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. 

The Ladder Theorem

 

Figure 1

I hadn't heard of the Ladder Theorem, shown in Figure 1, until I was confronted with the problem shown in Figure 2. It appeared in the thumbnail of a YouTube video and the problem was to find the value of \(x\).

Figure 2: screenshot from YouTube video

The problem looks deceptively easy but, unless you're familiar with the theorem, it's not. Using the theorem, the solution can be reached easily. Let's refer to Figure 1 (and ignore the \(x\) in Figure 2) and so we can see that \(x=4, y=10 \text{ and } z=5\). Thus \(x+y=14 \text{ and } y+z=15\). This leads to (and now the \(x\) refers to the \(x\) in Figure 2):$$
\begin{align} \frac{1}{x+19}+\frac{1}{10}&=\frac{1}{14}+\frac{1}{15}\\
\frac{1}{x+19}&=\frac{1}{14}+\frac{1}{15}-\frac{1}{10}\\&=\frac{15+14-21}{210}\\
&=\frac{8}{210}\\&=\frac{4}{105}\\x+19&=26.25\\x&=7.25 \end{align}$$A rather lengthy proof of the Ladder Theorem can by found by watching this YouTube video. It would seem that this theorem is part of a family of related theorems including:

• Ceva's Theorem
• Menelaus's Theorem
• Van Aubel's Theorem
• Steward's Theorem

I won't go into these other theorems in this post but this video discusses all of them.

Lines and Triangles

Consider the following problem as shown in Figure 1.

Figure 1

I just stumbled upon this problem, or rather a generalisation of this problem, and thought its solution was worthy of a mention. The solution was quite simple once I read it. Let's first review what the diagram shows. There are three parallel lines (labelled 1, 2 and 3) and another five non-parallel lines (labelled 4 to 8) that intersect with each other and also with the parallel lines. No three lines pass through a single point. How many triangles are formed?

Let's break the problem into two parts. Firstly let's deal with the five non-parallel lines. It takes three lines to make a triangle and so there are$$ \binom 5 3 = \binom 5 2 =\frac{5 \times 4}{2 \times 1}= 10 \text { possible combinations}$$So the five non-parallel lines form 10 triangles with each other. The three parallel lines can each be linked to a pair of non-parallel lines. This becomes:$$ \binom 5 2 \times 3 = 10 \times 3 = 30 \text{ possible combinations}$$So in total there are \(10 + 30 = 40\) triangles formed.

Generalising this, if there are \( n \) non-parallel lines and \(m\) parallel lines then:$$\text {The number of triangles formed is given by } \binom n 3 + \binom n 2 \times m$$Of course if none of the lines are parallel then \(m=0\) and the expression simplifies accordingly. It's interesting to count the number of points as well (shown in Figure 2):

Figure 2

The intersection of the lines produces 25 points, which I dutifully counted and labelled. If the points are taken in pairs, then each pair will generate a line with the result that there will be \( \binom {25} 2 = \frac {25 \times 24}{2 \times 1}=300\) lines which is clearly not the case because many of the lines are the same as others due to the large number of collinear points. 

However it can be noted that when two lines intersect, a point is created and so the following expression arises:$$ \binom 5 2 + 5 \times 3 = 10 + 15 =25 \text { which confirms my manual count}$$This can be generalised to:$$ \text {The number of points is given by }\binom n 2 + n \times m$$

Triangles with Optimal Dynamics

In my previous post, I explored the close-to-equilateral integer triangles and today I was reading another mathematical article about triangles, so-called triangles with optimal dynamics. I thought the topic worthy of a blog post. Here is a quote from the article:

When you set a ball in motion on a billiard table, it may seem as if anything is possible, but when a table has optimal dynamics, only two things truly are. The first is complete chaos, which is to say that the ball’s path will cover the entire table as time wears on. The second is periodicity — a repeating path like a ball pinging back and forth between two sides.

In tables without optimal dynamics, a wider range of possibilities exists, which makes the full analysis of all possible paths impossible: A ball could end up bouncing chaotically in one part of the table forever, never retracing its path, but also never covering the whole table.

What mathematicians do know is that there are at least eight kinds of triangles with optimal dynamics; the first was discovered in 1989 and the last in 2013. Whether there are more is anyone’s guess. 

 Let's look at each in turn:

• 1 : 1 : n means isosceles triangles with angles 10°, 10°, 160° or 20°, 20°, 140° etc.
  
• 1 : 2 : n means triangles with angles 10°, 20°, 150° or 20°, 40°, 120° etc.
  
• 3 : 4 : 5 represents a triangle with angles of 45°, 60° and 75°
  
• 2 : 3 : 4 represents a triangle with angles of 40°, 60° and 80°
  
• 3 : 5 : 7 represents a triangle with angles of 36°, 60° and 84°
  
• 1 : 4 : 7 represents a triangle with angles of 15°, 60° and 105°
  
• 2 : (n-2) : n represents a right angled triangles with angles of 2°, 88° or 90°, 4°, 86° or 6°, 84°, 90° or 10°, 80°, 90° or 12°, 78°, 90° or 18°, 72°, 90° or 20°, 70°, 90° or 30°, 60°, 90° or 36°, 54°, 90°
  
• 2 : n : n represents triangles like 2°, 89°,89° or 4°, 88°, 88° or 6°, 86°, 86° etc.

The article goes on to identify two quadrilaterals with optimal dynamics (shown below):