The Reuleaux Triangle Revisited

In all my posts since the middle of 2015, I've only mentioned the Reuleaux (pronounciation) Triangle once and there's a good reason for this. Let's look at my original post from the 9th of November 2021 titled Reuleaux Triangle. I wrote (text in blue):

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 this number stands in splendid isolation from its neighbours: 17358 (\(n\) = 9) and 40590 (\(n \) = 11). The former occurred before I started monitoring my diurnal age and the second will occur long after I'm dead. In my post, I included the image shown in Figure 1.

Figure 1

The post I made back then is a good one, if I do say so myself, and delves into some interesting properties and applications of the Reuleaux triangle. The reason the triangle has popped up again has to do again with my diurnal age (27336) but this time it's regions rather than vertices that are involved. The number is a member of OEIS A340639:

A340639

The number of regions inside 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:

1, 24, 145, 516, 1432, 3084, 6106, 10638, 17764, 27336, 41233, 58902, 82675, 111864, 149497, 194430, 250534, 316020, 395728, 487122, 596434, 720162, 865321, 1027974, 1216291, 1425348, 1664539, 1928022, 2226658, 2553204, 2920378, 3319536, 3764848, 4246638, 4780489, 5355414, 5988973 

As with the number mentioned in the earlier post, 27336 corresponds to the case of \(n\) = 10 and once again its neighbours (17764 and 41233) are far removed. The OEIS comments provide a link to a very colourful image of the regions. See Figure 2.

Figure 2: source

However, the regions can be seen more clearly perhaps in the case of \(n\) = 2. See Figure 3.

Figure 3: source

So it may be some time before the Reuleaux triangle gets a mention again because the only integers associated with it seem to relate to vertices and regions arising from the division of its sides into \(n\) equal parts.

Happy Triangular Numbers

I have to confess to treating triangular numbers with some complacency over the years. Let's recall that triangular numbers are of the form:$$ \frac{n \, (n+1)}{2} \text{ with } n \geq 1$$The first triangular numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431

Figure 1: Triangular numbers appear in Pascal's Triangle.
In fact 3rd diagonal of Pascal's Triangle, gives all triangular numbers as shown

Similarly I've grown rather complacent about happy numbers that have the property that repeated iterations of the sum of digits squared lead to 1. For example:$$ 27261 \rightarrow 94, 97, 130, 10, 1$$The reason that I chose 27261 is that it's the number associated with my diurnal age today.

It turns out however, that numbers that are both triangular and happy are rather rare. These types of numbers comprise OEIS A076712 and the initial members of the sequence are:

1, 10, 28, 91, 190, 496, 820, 946, 1128, 1275, 2080, 2211, 2485, 3321, 4278, 8128, 8256, 8778, 9591, 9730, 11476, 12090, 12880, 13203, 13366, 13530, 15753, 16471, 17205, 17578, 20910, 21115, 21321, 22791, 24753, 25651, 27261, 29890, 30135, 31626, 33670, 35245

As can be seen, it will be quite some time before I meet the next such number: 29890. The main reason for creating this post was to draw attention to an interesting website titled Fascinating Triangular Numbers run by Shyam Sunder Gupta. It was begun on October 26th 2002 but remains active. There is a plethora of information on this site about various properties of triangular numbers so it's a wonderful resource.

What follows are just two examples from the site:

Example 1: The only known example of a Pythagorean triangle (\(a, b, c\)) where \(a, b, c\) are triangular numbers is (8778, 10296, 13530): $$ \begin{align} 8778^2 + 10296^2 &= 13530^2 \\ (\text{T}_{132})^2 + (\text{T}_{143})^2 &= (\text{T}_{164})^2 \end{align} $$Example 2: The only known examples of a Pythagorean triangle such that both Perimeter as well as Area are triangular numbers are: $$ \begin{align} &3312, 14091, 14475) \\ \text{with Perimeter } &= 31878 = T_{252} \\ \text{and Area } &= 23334696 = T_{6831}\\  \\&3405996, 8013265, 8707079 \\ \text{with Perimeter } &= 20126340 = T_{6344} \\ \text{and Area } &= 13646574268470 = T_{5224284} \end{align} $$

Rectangles and Squares

If we envisage a semiprime that is not a square number as a rectangle then a number like 15 that is equal to 3 x 5 could be represented as shown in Figure 1:

Figure 1: created using Geoboard

The average of 3 and 5 is 4 and a 4 x 4 square has the same perimeter as the 3 x 5 rectangle. Both are 16 units in perimeter. See Figure 2.

Figure 2: created using Geoboard

Though the rectangle and the square have the same perimeter, they have different areas. The rectangle has an area of 15 square units and the square has an area of 16 square units. 16 is a square number and the semiprime 15 is linked to it via its two prime factors:$$\frac{3 +5}{2} \times 4 = 16$$The square number divided by 4 gives the side of the associated square. Not every semiprime can be linked to square number in this way. Take 33 with prime factors of 3 and 11 as an example:$$\frac{3 +11}{2} \times 4 = 28$$In general, if a semiprime has two distinct prime factors \(a\) and \(b\), then the condition is that \(2 \times (a+b) \) needs to be a square number. 

In the range up to 40,000, only 172 of the 9790 semiprimes qualify (permalink). Here is the list:

15, 65, 77, 87, 141, 247, 301, 335, 481, 589, 591, 671, 717, 767, 785, 1007, 1167, 1247, 1271, 1351, 1415, 1501, 1527, 1661, 1937, 1967, 2071, 2077, 2157, 2257, 2317, 2391, 2977, 3007, 3047, 3101, 3197, 3215, 3439, 3997, 4061, 4087, 4237, 4385, 4487, 4607, 4829, 4927, 5111, 5777, 6031, 6077, 6161, 6487, 6497, 6541, 6557, 6751, 6927, 7087, 7265, 7341, 7357, 7361, 7967, 8189, 8479, 8557, 9217, 9271, 9287, 9517, 9991, 10077, 10157, 10231, 10727, 11041, 11327, 12209, 12687, 12877, 12989, 13511, 13847, 14317, 14397, 15007, 15185, 15917, 16081, 16397, 16769, 16897, 16957, 17711, 17951, 18141, 18157, 18527, 18807, 19117, 19127, 19367, 19511, 19679, 19741, 19757, 20017, 20191, 20567, 20687, 20711, 20877, 21041, 21421, 21697, 23015, 23231, 23377, 23389, 23729, 23839, 24727, 24737, 24887, 24961, 25341, 25661, 25837, 25967, 25985, 26797, 26909, 27341, 27661, 28247, 28417, 29047, 29135, 29431, 30237, 30311, 30461, 30847, 31597, 31681, 32047, 32551, 32567, 32847, 33527, 34207, 34241, 34647, 34951, 35249, 35741, 35807, 36077, 36391, 36737, 37327, 37437, 37777, 38081, 38191, 38407, 38551, 38687, 39421, 39665

Let's test the second member of the sequence, 65, with factors of 5 and 13. We see that:$$2 \times (5+13)=36$$The associated square has a side of 9 units. However, different semiprimes can produce the same square number. Take the semiprime 77 with prime factors of 7 and 11 as an example: $$2 \times (7+11)=36$$See Figure 3 where the two rectangles associated with the two different semiprimes are shown together with the associated square.

Figure 3: created using Geoboard

If we want to work backwards from the square numbers to the semiprimes, then it's a question of dividing the square number by 2 and partitioning the resultant number into two parts such that each is prime. The results (permalink) are shown in the table below with only those semiprimes up to 40,000 displayed. The algorithm is easily modified so as to remove this filter and show all semiprimes associated with square numbers up and including 40,000.

square   half-square   rectangle    semiprime

  16       8             [5, 3]       15
  36       18            [13, 5]      65
  36       18            [11, 7]      77
  64       32            [29, 3]      87
  64       32            [19, 13]     247
  100      50            [47, 3]      141
  100      50            [43, 7]      301
  100      50            [37, 13]     481
  100      50            [31, 19]     589
  144      72            [67, 5]      335
  144      72            [61, 11]     671
  144      72            [59, 13]     767
  144      72            [53, 19]     1007
  144      72            [43, 29]     1247
  144      72            [41, 31]     1271
  196      98            [79, 19]     1501
  196      98            [67, 31]     2077
  196      98            [61, 37]     2257
  256      128           [109, 19]    2071
  256      128           [97, 31]     3007
  256      128           [67, 61]     4087
  324      162           [157, 5]     785
  324      162           [151, 11]    1661
  324      162           [149, 13]    1937
  324      162           [139, 23]    3197
  324      162           [131, 31]    4061
  324      162           [109, 53]    5777
  324      162           [103, 59]    6077
  324      162           [101, 61]    6161
  324      162           [89, 73]     6497
  324      162           [83, 79]     6557
  400      200           [197, 3]     591
  400      200           [193, 7]     1351
  400      200           [181, 19]    3439
  400      200           [163, 37]    6031
  400      200           [157, 43]    6751
  400      200           [139, 61]    8479
  400      200           [127, 73]    9271
  400      200           [103, 97]    9991
  484      242           [239, 3]     717
  484      242           [229, 13]    2977
  484      242           [223, 19]    4237
  484      242           [211, 31]    6541
  484      242           [199, 43]    8557
  484      242           [181, 61]    11041
  484      242           [163, 79]    12877
  484      242           [139, 103]   14317
  576      288           [283, 5]     1415
  576      288           [281, 7]     1967
  576      288           [277, 11]    3047
  576      288           [271, 17]    4607
  576      288           [269, 19]    5111
  576      288           [257, 31]    7967
  576      288           [251, 37]    9287
  576      288           [241, 47]    11327
  576      288           [229, 59]    13511
  576      288           [227, 61]    13847
  576      288           [199, 89]    17711
  576      288           [191, 97]    18527
  576      288           [181, 107]   19367
  576      288           [179, 109]   19511
  576      288           [157, 131]   20567
  576      288           [151, 137]   20687
  576      288           [149, 139]   20711
  676      338           [331, 7]     2317
  676      338           [307, 31]    9517
  676      338           [277, 61]    16897
  676      338           [271, 67]    18157
  676      338           [241, 97]    23377
  676      338           [229, 109]   24961
  676      338           [211, 127]   26797
  676      338           [199, 139]   27661
  676      338           [181, 157]   28417
  784      392           [389, 3]     1167
  784      392           [379, 13]    4927
  784      392           [373, 19]    7087
  784      392           [349, 43]    15007
  784      392           [331, 61]    20191
  784      392           [313, 79]    24727
  784      392           [283, 109]   30847
  784      392           [241, 151]   36391
  784      392           [229, 163]   37327
  784      392           [211, 181]   38191
  784      392           [199, 193]   38407
  900      450           [443, 7]     3101
  900      450           [439, 11]    4829
  900      450           [433, 17]    7361
  900      450           [431, 19]    8189
  900      450           [421, 29]    12209
  900      450           [419, 31]    12989
  900      450           [409, 41]    16769
  900      450           [397, 53]    21041
  900      450           [389, 61]    23729
  900      450           [383, 67]    25661
  900      450           [379, 71]    26909
  900      450           [367, 83]    30461
  900      450           [353, 97]    34241
  900      450           [349, 101]   35249
  900      450           [347, 103]   35741
  900      450           [337, 113]   38081
  1024     512           [509, 3]     1527
  1024     512           [499, 13]    6487
  1024     512           [439, 73]    32047
  1024     512           [433, 79]    34207
  1156     578           [571, 7]     3997
  1156     578           [547, 31]    16957
  1156     578           [541, 37]    20017
  1156     578           [499, 79]    39421
  1296     648           [643, 5]     3215
  1296     648           [641, 7]     4487
  1296     648           [631, 17]    10727
  1296     648           [619, 29]    17951
  1296     648           [617, 31]    19127
  1296     648           [607, 41]    24887
  1296     648           [601, 47]    28247
  1296     648           [587, 61]    35807
  1444     722           [719, 3]     2157
  1444     722           [709, 13]    9217
  1444     722           [691, 31]    21421
  1600     800           [797, 3]     2391
  1600     800           [787, 13]    10231
  1600     800           [769, 31]    23839
  1600     800           [757, 43]    32551
  1764     882           [877, 5]     4385
  1764     882           [863, 19]    16397
  1764     882           [859, 23]    19757
  1764     882           [853, 29]    24737
  1764     882           [839, 43]    36077
  1936     968           [937, 31]    29047
  2116     1058          [1051, 7]    7357
  2116     1058          [1039, 19]   19741
  2116     1058          [1021, 37]   37777
  2304     1152          [1129, 23]   25967
  2304     1152          [1123, 29]   32567
  2500     1250          [1237, 13]   16081
  2500     1250          [1231, 19]   23389
  2916     1458          [1453, 5]    7265
  2916     1458          [1451, 7]    10157
  2916     1458          [1447, 11]   15917
  2916     1458          [1439, 19]   27341
  3136     1568          [1549, 19]   29431
  3364     1682          [1669, 13]   21697
  3364     1682          [1663, 19]   31597
  3600     1800          [1789, 11]   19679
  3600     1800          [1787, 13]   23231
  3600     1800          [1783, 17]   30311
  4096     2048          [2029, 19]   38551
  4356     2178          [2161, 17]   36737
  4624     2312          [2309, 3]    6927
  4900     2450          [2447, 3]    7341
  4900     2450          [2437, 13]   31681
  5184     2592          [2579, 13]   33527
  5476     2738          [2731, 7]    19117
  6084     3042          [3037, 5]    15185
  6724     3362          [3359, 3]    10077
  7056     3528          [3517, 11]   38687
  7396     3698          [3691, 7]    25837
  8464     4232          [4229, 3]    12687
  9216     4608          [4603, 5]    23015
  9604     4802          [4799, 3]    14397
  10000    5000          [4993, 7]    34951
  10404    5202          [5197, 5]    25985
  11664    5832          [5827, 5]    29135
  12100    6050          [6047, 3]    18141
  12544    6272          [6269, 3]    18807
  13924    6962          [6959, 3]    20877
  15876    7938          [7933, 5]    39665
  16900    8450          [8447, 3]    25341
  20164    10082         [10079, 3]   30237
  21904    10952         [10949, 3]   32847
  23104    11552         [11549, 3]   34647
  24964    12482         [12479, 3]   37437

One could extend this idea to sphenic numbers and three dimensions. Each sphenic number can be interpreted as a brick or rectangular prism. What sphenic numbers have surface areas that are the same as that of cubes with integer sides? The list of such sphenic numbers is shown in the table below (permalink):

sphenic   factors         SA      cube side   SA

  374       2 * 11 * 17     486     9           486
  710       2 * 5 * 71      1014    13          1014
  3110      2 * 5 * 311     4374    27          4374
  3590      2 * 5 * 359     5046    29          5046
  4454      2 * 17 * 131    5046    29          5046
  6182      2 * 11 * 281    7350    35          7350
  7190      2 * 5 * 719     10086   41          10086
  8911      7 * 19 * 67     3750    25          3750
  9494      2 * 47 * 101    10086   41          10086
  10502     2 * 59 * 89     11094   43          11094
  10507     7 * 19 * 79     4374    27          4374
  11798     2 * 17 * 347    13254   47          13254
  18518     2 * 47 * 197    19494   57          19494
  18854     2 * 11 * 857    22326   61          22326
  20390     2 * 5 * 2039    28566   69          28566
  24134     2 * 11 * 1097   28566   69          28566
  27559     7 * 31 * 127    10086   41          10086

The T-square Fractal

Figure 1: source

Yesterday (when I was young) I came across the T-square fractal in the context of turning 26598 days old. Figure 1 shows a screenshot of my Twitter tweet for the day. The OEIS entry runs as follows:

A227621

The nearest integer of perimeter of T-square (fractal) after n-iterations, starting with a unit square.

The initial members of the sequence are:

4, 8, 14, 23, 37, 57, 87, 133, 201, 304, 457, 688, 1034, 1553, 2331, 3499, 5251, 7878, 11819, 17731, 26598, 39899, 59851, 89778, 134669, 202005, 303010, 454517, 681778, 1022668, 1534004, 2301009, 3451515, 5177275, 7765914

Figure 2, taken from the Wikipedia entry, shows the initial steps in the creation of this fractal:

Figure 2: source

Figure 3 shows further details of the process:

Figure 3: source

Starting from an initial unit square, the fractal is bounded by the square with a side of two units since:$$ \sum_{n=0}^{\infty} \frac{1}{\,2^n}=2$$As the area of the fractal gets closer and closer to 2, the perimeter gets longer and longer. After 20 iterations, the perimeter is 26598 units in length. As this source explains:

The fractal dimension is the ratio between the "size" of the object and its "area". For example, a simple area has a fractal dimension of 2, which means that if you make it growth by \(x\), its area will be multiplied by \(x^2\). A fractal with a dimension of 1.5 will have its area multiplied by \(x^{1.5}\).

In the case of the T-Square, it has a dimension of 2 because its area nearly entirely fills the inner space within it. However, the dimension of its boundary is \( \frac{\log 3}{\log 2} \approx 1.58\).

OnlineMathTools enables the creation of a variety of fractals, including the T-square fractal, that can be customised in various ways. Figure 4 shows an example of such a fractal.

Figure 4: T-square fractal after 4 iterations