Infinities – According to The Kidd

INFINITIES 12 – Infinities Left of the Decimal

This is the twelfth in a series of posts exploring infinity, entities and ideas infinite, infinities larger than other infinities, infinity paradoxes, infinite perimeters, infinite regress and infinite sequences such as cyclic numbers.

In the previous post it was pointed out that an infinite string of nines to the right of the decimal (0.9999999 . . .) is equal to one but it was never proved. This post will start with a simple proof of that fact. Then there will be a demonstration of what an infinite string of nines to the LEFT of the decimal is equal to. The answer is ridiculous.

Let k = 0.9999999 . . .
10k = 9.9999999 . . .
10k – k = 9
9k = 9
k = 1 = 0.9999999 . . .
Now, let . . . 999999.0 = m
Equation 1: . . . 999999 = m
Equation 2: . . . 9999990.0 = 10m
(. . . 999999) – (. . . 9999990) = m – 10m
9 = -9m
m = -1
. . . 999999.0 = -1

To confirm this bizarre result, add 1 to m and if m is equal to -1 then adding 1 should give us zero. What is . . . 999999 plus 1 equal to? 9 +1 = 0, carry the 1 into the tens column, 1 +9 there equals 10 again, place the zero next to the first zero down below and carry the 1 into the hundreds column, 1 + 9 there equals 10 again, place another zero next to the two zeroes in the answer at the bottom and carry the 1 into the thousands column. Do this forever and you get an infinite string of zeroes. So, apparently, . . . 999999 = -1 . If we subtract 6 from each side, we get that . . . 999993 = -7, and if we subtract 40 we get that . . . 999959 = -41 and so on.

Now, consider what the following infinitely large number is equal to (this is a number consisting of an endlessly repeating chain of the six digits 857142, plus 1):

. . . 857142 857142 857143.0

What happens if you multiply this number by 7?

. . . 857142 857142 857143 x 7 = . . . 000000000000000001, i.e. simply 1

Therefore . . . 857142 857142 857143.0 = 1 / 7.

What do you suppose the infinitely large number . . . 666667 is equal to? If you multiply it by 3 you get 1, so . . . 666667 must equal 1 / 3.

What do you suppose . . . 888889 is equal to?

To find the negative of one of these numbers, you can either multiply it by -1 (which can be tricky), or you can find the number’s complement and add one to that complement. Two numbers are complements if they add up to nine. For example, the complement of 8 is 1, the complement of 4 is 5. The complement of . . . 857142857143 is . . . 142857142856. Adding 1 gives you . . . 142857142857 which theoretically equals – 1 / 7. How can we verify that? Since . . . 857142857143 equals 1 / 7 and . . . 142857142857 equals – 1 / 7, then if we add them together they should equal zero – and that’s exactly what does happen.

PIERRE DE FERMAT
By Unknown author – https://web.archive.org/web/20191028044928/http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Fermat.html, Public Domain, https://commons.wikimedia.org/w/index.php?curid=36804

Andrew Wiles used a prime subset of numbers like these to solve Fermat’s Last Theorem, an infamous theorem which had gone unsolved for 358 years despite the fact that the greatest mathematicians of the last 358 years had attempted to prove it, all unsuccessfully. For more information on these numbers – https://www.youtube.com/watch?v=tRaq4aYPzCc&ab_channel=Veritasium

By the way, . . . 888888889 = 1 / 9 . To confirm this multiply the left side by 9 and see if you get 1.

PHILOSOPHICAL PERPLEXITIES ABOUT INFINITY

There are some Christian apologists who apparently are confused about what infinity is all about. A particularly prominent theist by the name of William Lane Craig gets infinity wrong in various ways in his defence of the Kalam Cosmological Argument for the existence of God. Similarly, the theist Alvin Plantinga’s defence of the Ontological Argument for the existence of God also rests on a misunderstanding of the nature of infinity.

William Lane Craig’s writings and speeches delve deeply into tensed and tenseless time, presentism and the nature of spacetime. He defends a neo-Lorentzian interpretation of Einstein’s Special Theory of Relativity, yet at the same time he is wrong about some basic mathematical concepts which I learned in high school. Craig also misses the point about Hilbert’s extraordinary Hotel Infinity (a topic dealt with in depth in the fourth post of this series of posts).

These philosophical treatments of infinity can become esoteric exercises in finitism and modal logic. I have gone into these matters in some detail at the end of this post if anyone is interested.

Post 1 – Infinity Everywhere – https://thekiddca.wordpress.com/2024/04/06/infinities-infinity-everywhere/

Post 2 – Snowflake Curve – https://thekiddca.wordpress.com/2024/04/13/infinities-2-theres-no-business-like-snow-business/

Post 3 – Ch’i Ch’iao T’u – https://thekiddca.wordpress.com/2024/04/20/infinities-3-infinitesimal-chi-chiao-tu/

Post 4 – Hotel Infinity – https://thekiddca.wordpress.com/2024/04/27/infinities-4-no-vacancies-but-rooms-still-available-at-hotel-infinity/

Post 5 – Pythagorean Infinity – https://thekiddca.wordpress.com/2024/05/04/infinities-5-pythagorean-infinity/

Post 6 – Rep-tiles 1 – https://thekiddca.wordpress.com/2024/05/12/infinities-6-welcome-to-the-rep-tile-house-part-1-of-5/

Post 7 – Rep-tiles 2 – https://thekiddca.wordpress.com/2024/05/18/infinities-7-welcome-to-the-rep-tile-house-part-2-of-5/

Post 8 – Rep-tiles 3 – https://thekiddca.wordpress.com/2024/05/25/infinities-8-welcome-to-the-rep-tile-house-part-3-of-5/

Post 9 – Rep-tiles 4 – https://thekiddca.wordpress.com/2024/06/01/infinities-9-welcome-to-the-rep-tile-house-part-4-of-5/

Post 10 – Rep-tiles 5 – https://thekiddca.wordpress.com/2024/06/08/infinities-10-welcome-to-the-rep-tile-house-part-5-of-5/

Post 11 – Cyclic Power –
– https://thekiddca.wordpress.com/2024/06/15/infinities-11-cyclic-power/

SOME PHILOSOPHICAL PERPLEXITIES – MORE DETAILS

William Lane Craig is very sure of himself and when experts in fields outside his area of expertise (e.g. Mathematics, Cosmology and Astrophysics) challenge him he doesn’t like it. He has said with some impatience that there could never be such a thing as the David Hilbert’s Hotel Infinity, missing Hilbert’s point completely. Hilbert came up with the idea of Hotel Infinity to demonstrate that one must be careful when dealing with the concept of infinity because if you’re not careful you end up with bizarre impossible entities like Hotel Infinity. Hotel Infinity has an infinite number of rooms, each one occupied, but if a new guest arrives that guest can be given a room of her own while all the remaining guests can still remain in rooms alone. A thousand new guests could also be accommodated. So could an infinite number of new guests. So could an infinite number of sets of infinite numbers of guests.

Cantor did pioneering work with transfinite numbers and someone properly schooled in Mathematics would be fine with Cantor’s work. However, some Christian apologists seem to be unable or unwilling to understand something established mathematically 140 years ago. I have also heard Christian apologists misrepresent how probability works (I have no cites so you can accept this observation or not).

William Lane Craig argues that the universe had a beginning, everything has a cause, and that the only cause for the universe that makes sense is the Christian God. This is known as the Kalam Cosmological Argument. The argument is based on Craig’s misunderstanding of the nature of infinity (as well as various unwarranted assumptions). At one point he refers to infinity as the limit of an infinite series. He doesn’t seem to comprehend that infinity is not a number, nor does he understand the difference between a converging and a diverging infinite series. High school math.

His arguments also involve tensed and tenseless time, presentism and the nature of spacetime. He believes that the universe has not existed for an infinite amount of time in the past. He is convinced that God led a timeless existence, then He created the universe which included everything, even time itself. He believes in what metaphysicists refer to as the A-theory of time, i.e. that there are tensed facts (e.g. “it is now dinner time”) which cannot be reduced to tenseless facts (e.g. “it is dinner time at six P.M. on June 22, 2024”). He seems to think of the present as an individual infinitesimal slice of existence sliding along (his phrase is “absolutely becoming”) between the past and the future. This is at odds with Einstein’s demonstration that there is no such thing as simultaneity.

Unfortunately empirical science little by little continues to undermine Craig’s assertions. He argues that one can never have something come into existence from nothing, a conclusion which has been proven wrong by the discovery of quantum fluctuations leading to sub-atomic particles popping in and out of existence. More recently the discoveries of gravitational waves is also at odds with Craig’s beliefs. In 2011 noted astrophysicist Lawrence Krauss went as far as to accuse Craig of “disingenuous distortions, simplifications, and outright lies”.

One of the people whose views William Lane Craig is enamoured of is a theist named Alvin Plantinga. In 1960 Plantinga vigorously defended something called The Ontological Argument for the existence of God. That particular argument was proposed by St. Anselm (1033 – 1109) almost a thousand years ago, and yet before the ink was dry on St. Anselm’s manuscript Gaunilo of Marmoutiers, a contemporary of St. Anselm, was able to show that it was flawed. The Ontological Argument was also rejected by Thomas Aquinas (1225 – 1274), David Hume (1711 – 1776) and Immanual Kant (1724 – 1804). Aquinas and Kant, may it be noted, were two of the six or seven greatest philosophers who ever lived (IMHO). More to the point, Plantinga’s arguments also rest on a misunderstanding of infinity.

ST. THOMAS AQUINAS
By Bartolomé Esteban Murillo – Own work Amuley, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=33176874

As Keith Backman demonstrated in 2016, Plantinga’s treatment of infinite magnitudes and maximal greatness in his ‘proof’ of the Ontological Argument are fallacious. Backman explains how both St. Anselm and Platinga “propose that attributes of infinite magnitude should be manipulated, augmented or aggregated so as to obtain an entity with a magnitude distinguishable from the infinitely great properties of its starting components” (to quote the summary on the back of the book). Plantinga’s argument rests on several fallacies. One of the fallacies involves an invalid treatment of characteristics involving implicitly infinite magnitude as if they possess a definite measure. Plantinga also groups disparate appreciations of God’s existential status (introduced as imaginary), but then treats as substantive for purposes of reasoned argument (for more details see Creating God From Nothing: Fallacy and Subterfuge in Ontological Arguments for the Existence of God, by Keith Backman, Bookstand Publishing, 2016, page 55).

Tread the terrain of the infinite with great care.

INFINITIES 11 – Cyclic Power

This is the eleventh in a series of posts exploring infinity, entities and ideas infinite, infinities larger than other infinities, infinity paradoxes, infinite perimeters, infinite regress and infinite sequences such as cyclic numbers.

Different people have different favourite numbers. When I was but a child my favourite number was four because that was the jersey number of Red Kelly who was a high achieving hockey player on the Toronto Maple Leafs back when they were winning Stanley Cups a lot. Now my favourite number is 142857. A lot has been written about this number but this post will, in all probability, reference some little known properties of the number.

What do you get if you multiply 142857 by 2? You get 285714. Notice anything unusual about that? Yes, the digits in both 142857 and 285714 are not only the same but they are also in the same order. It’s just that they start with different digits. What do you think will happen if you multiply 142857 by 3? Here is what you get if you multiply 142857 by all of the digits from one to six:

142857 x 1 = 142857
142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142

Now what? We’ve generated six results using all six digits in 142857 as the leading digits in the answers, so what happens if we multiply 142857 by seven? Any guesses? 142857 is called a cyclic number and people have known about such numbers for quite awhile. Lewis Carroll, who is most famous for writing Alice in Wonderland and Through the Looking Glass, was also a mathematician and a pioneer in the field of symbolic logic. He was analysing cyclic numbers a century and a half ago.

Curiously, all of the cyclic numbers except 142857 have leading zeroes. For example, the next larger cyclic number is 0588235294117647 so we have

1 x 0588235294117647 = 0588235294117647
2 x 0588235294117647 = 1176470588235294
3 x 0588235294117647 = 1764705882352941
. . .
16 x 0588235294117647 = 9411764705882352

What do you get if you calculate the decimal expression of one seventh? You get 0.142857 142857 142857 . . . and that same group of digits, 142857, keeps repeating forever. You can guess what two sevenths, three sevenths and so on look like in decimal form:

1 / 7 = 0.142857 142857 142857 . . .
2 / 7 = 0.285714 285714 285714 . . .
3 / 7 = 0.428571 428571 428571 . . .
4 / 7 = 0.571428 571428 571428 . . .
5 / 7 = 0.714285 714285 714285 . . .
6 / 7 = 0.857142 857142 857142 . . .

But then when you multiply 142857 by 7 you get 999999 (a special case of Midy’s Theorem). So what does seven sevenths look like? It must equal 0.999999 999999 999999 and so on. But seven sevenths equals one. Therefore 1 = 0.999999 999999 999999 . . . After all 0.9999 . . . is equal to the infinite series nine tenths plus 9 hundredths plus nine thousandths plus nine ten thousandths and so on, and if you keep adding together these smaller and smaller amounts you get closer and closer to one. If you were able to keep adding these numbers forever (which is, of course, impossible) you would get to one. So we say that the limit of all these fractions is one, and a limit is something that you can get as close to as you like but you can never reach. This is a fundamental concept used in calculus.

Looking at the decimal expression of one seventh also helps to explain the cyclic property of 142857. Take a look at this:

1/7 = 0.142857 142857 142857 . . .

Multiply both sides by 10. 10/7 = 1.428571 428571 428571 . . .

10/7 = 7 / 7 + 3 / 7 = 1 + 3 / 7 = 1.428571 428571 428571 . . .

So 3/7 = 0.428571 428571 428571 . . .

Multiply by 10 again. 30/7 = 4.285714 285714 285714 . . .

30/7 = 28/7 + 2/7 = 4 + 2/7 = 4.285714 285714 285714 . . .

So 2/7 = 0.285714 285714 285714 . . . and so on.

999999 is also equal to 1000000 – 1, that is 10⁶ – 1. Here are the first five cyclic numbers expressed using exponents:

(10⁶ – 1) / 7 = 142857 (6 digits)
(10¹⁶ – 1) / 17 = 0588235294117647 (16 digits)
(10¹⁸ – 1) / 19 = 052631578947368421 (18 digits)
(10²² – 1) / 23 = 0434782608695652173913 (22 digits)
(10²⁸ – 1) / 29 = 0344827586206896551724137931 (28 digits)

The generators of the first five cyclic numbers are 7, 17, 19, 23 and 29. These are all prime numbers. All cyclic numbers are also prime numbers but not all prime numbers are cyclic numbers. However we do know that 37.395 . . . % of all prime numbers are cyclic numbers. It would seem that cyclic numbers, like prime numbers, are infinite. There are also other pseudocyclic numbers whose digit sequence remains intact while the leading digit changes. However their multiples are not consecutive integers so they are not genuine cyclic numbers.
For example – 076923 x 1 = 076923, 076923 x 3 = 230769, 076923 x 4 = 307692, 076923 x 9 = 692307, 076923 x 10 = 769230, 076923 x 12 = 923076 .

Cyclic and pseudocyclic numbers can be generated by the expression (b^{p – 1} – 1) / p where b is the base being used and p is a prime number greater than 3 which does not divide evenly into b. Therefore, if b equals 10 (using the decimal system, i.e. base 10), and p = 7, this expression becomes (10^{7 – 1} – 1) / 7 = (10⁶ – 1) / 7 = 999 999 / 7 = 142857.

Sometimes it’s interesting just exploring number patterns, and then trying to figure out how the pattern is generated. The following pattern is sort of regular but isn’t quite, and I have no idea if it is significant:

Divide the six-digit number 142857 into three two-digit parts and add them up (2 goes into 6 evenly): 14 + 28 + 57 = 99. 3 TWO-DIGIT NUMBERS – TWO NINES.
Divide the six-digit number 142857 into two three-digit parts and add them up (3 goes into 6 evenly): 142 + 857 = 999. 2 THREE-DIGIT NUMBERS – THREE NINES.
If you try to divide the six-digit number 142857 into four-digit parts you can’t do it since four doesn’t go into six evenly so instead divide the 12-digit number 142857142857 into four-digit parts, i.e. three of them (4 goes into 12 evenly, three times) and add those four-digit parts up: 1428 + 5714 +2857 = 9999. 3 FOUR-DIGIT NUMBERS – FOUR NINES.
Divide the 30-digit number 142857142857142857142857142857 into six five-digit parts (5 doesn’t go evenly into 6, 12, 18 or 24, but it does go evenly into 30) and add them up: 14285 + 71428 + 57142 + 85714 + 28571 + 42857 = 299997. This is not the a string of nines, but at least 2 + 7 equals 9, so we sort of have five nines. 6 FIVE-DIGIT NUMBERS – FIVE NINES (SORT OF).
Divide the 36-digit number 142857142857142857142857142857142857 into six six-digit parts (6 goes into 36 evenly) and add them up: 142857 + 142857 + 142857 + 142857 + 142857 + 142857 = 999999. 6 SIX-DIGIT NUMBERS – SIX NINES.
Divide the 42-digit number 142857142857142857142857142857142857142857 into six seven-digit parts (7 doesn’t go evenly into 6, 12, 18, 24, 30 or 36, but it does go evenly into 42) and add them up: 1428571 + 4285714 + 2857142 + 8571428 + 5714285 + 7142857 = 29999997, and again 2 + 7 = 9 so we sort of have seven nines. 6 SEVEN-DIGIT NUMBERS – SEVEN NINES (SORT OF).
Divide the 24-digit number 142857142857142857142857 into three eight-digit parts (8 doesn’t go evenly into 6, 12 or 18, but it does go evenly into 24) and add them up: 14285714 + 28571428 + 57142857 = 99999999. 3 EIGHT-DIGIT NUMBERS = EIGHT NINES.
Divide the 18-digit number 142857142857142857 into two nine-digit parts (9 doesn’t go evenly into 6 or 12, but it does go evenly into 18) and add them up: 142857142 + 857142857 = 999999999. 2 NINE-DIGIT NUMBERS = NINE NINES.

Does this work for ten-digit parts?

The cyclic number 142857 contains the digits from 1 to 9 except for all the multiples of three. I wonder if that is significant? What do you get if you take the last three digits of 142857 and square it, and subtract the square of the first three digits? That is (857)² – (142)² = ?

Finally, here is a curious cyclic-related patterns which I stumbled across in a fascinating 2020 paper by Santanu Bandyopadhyay published in Mumbai, India – https://www.ese.iitb.ac.in/~santanu/RM3.pdf :

What happens if you insert 9 into the middle of 142857 giving you the seven-digit number 1429857, then multiply that number by the numbers from one to six? –

1429857 x 1 = 1429857
1429857 x 2 = 2859714
1429857 x 3 = 4289571
1429857 x 4 = 5719428
1429857 x 5 = 7149285
1429857 x 6 = 8579142

What happens if you insert two 9’s? I wonder why these things happen?

Next post: INFINITIES LEFT AND RIGHT

Post 1 – https://thekiddca.wordpress.com/2024/04/06/infinities-infinity-everywhere/

Post 2 – https://thekiddca.wordpress.com/2024/04/13/infinities-2-theres-no-business-like-snow-business/

Post 3 – https://thekiddca.wordpress.com/2024/04/20/infinities-3-infinitesimal-chi-chiao-tu/

Post 4 – https://thekiddca.wordpress.com/2024/04/27/infinities-4-no-vacancies-but-rooms-still-available-at-hotel-infinity/

Post 5 – https://thekiddca.wordpress.com/2024/05/04/infinities-5-pythagorean-infinity/

Post 6 – https://thekiddca.wordpress.com/2024/05/12/infinities-6-welcome-to-the-rep-tile-house-part-1-of-5/

Post 7 – https://thekiddca.wordpress.com/2024/05/18/infinities-7-welcome-to-the-rep-tile-house-part-2-of-5/

Post 8 – https://thekiddca.wordpress.com/2024/05/25/infinities-8-welcome-to-the-rep-tile-house-part-3-of-5/

Post 9 – https://thekiddca.wordpress.com/2024/06/01/infinities-9-welcome-to-the-rep-tile-house-part-4-of-5/

Post 10 –

INFINITIES 10 – Welcome to the Rep-tile House Part 5 of 5

This is the tenth in a series of posts exploring infinity, entities and ideas infinite, infinities larger than other infinities, infinity paradoxes, infinite perimeters, infinite regress and infinite sequences such as cyclic numbers.

Once again, this post features rep-tiles. A rep-tile (e.g. a square) is a shape which can be divided into smaller copies of itself, and all those smaller copies are the same size.

This first illustration features two varieties of rep-tile. The first variety can be seen in the upper left hand corner. A second copy of the same thing can be seen in the lower right hand corner as well:

This rep-tile is made up of nine squares but at the same time four of them can be positioned together to form a perfect square. Therefore this rep-tile is made up of thirty-six equal-sized smaller replications of itself, as pictured. In the upper right hand corner is a second variety of rep-tile, similar to but not the same as the first variety:

A second copy of this variety is also visible in the lower left hand corner. Like the first variety, this rep-tile is made up of nine squares and at the same time four of them can be positioned together to form a perfect square. Therefore this rep-tile, like the first variety, is made up of thirty-six equal-sized smaller replications of itself, as pictured.

There is also an interesting design in the centre:

Here we have four identical crosses. Each cross is divided up into geometric figures called pentominoes. A pentomino is a shape consisting of five connected squares. Five squares can be joined together in twelve different ways. Only nine of those twelve are used in each of these crosses. Before continuing you might want to see if you can figure out what the other three pentominoes are. Here is my illustration of pentominoes within pentominoes:

The twelve black shapes are the twelve different pentominoes. There are three giant pentominoes here forming a giant rectangle, and that rectangle is also divided up into the twelve pentominoes separated by white lines. Each of those twelve are divided into nine smaller pentominoes. Each of those nine are in turn divided into nine smaller pentominoes. The design at the bottom also consists of pentominoes. Here is another work made up of complex groupings of pentominoes:

In this next work I have shown how 12 pentominoes (five squares each) are equal in area to the cross (six squares):

Around the main central figure here is a white frame also made up of pentominoes (and blue circle arcs). Around the outside of that frame is a collection of smaller pentominoes. The panel at the bottom is a demonstration of how the twelve pentominoes can be arranged to form a rectangle in two different ways. In this next work there are brightly covered pentominoes surrounding a blue staircase. That staircase is composed of hexominoes (a hexomino is a shape made up of six squares):

Finally, here is another cross made up of pentominoes and around the outside there are multi-coloured hexominoes.

Back to the rep-tiles, here we have one rep-tile in two different sizes and eight different colours:

This rep-tile is made up of nine squares again, basically a rectangle (two squares by four squares) plus a single ninth square projecting out on one side. Once again, four of these rep-tiles can be positioned to form a square. Therefore each rep-tile can be divided up into thirty-six (four times nine) smaller equal-sized rep-tiles identical to the large rep-tile. There is a large version of this on the left side comprised of sharp somewhat dark colours (except for the orange):

Two more smaller versions of this version can be found in the middle. The second version is made up of lighter colours (except for the blue) on the right side, also a large rep-tile consisting of thirty-six smaller rep-tiles:

Two slightly smaller duplicates of this second version can also be seen in the centre, for a total of four rep-tiles in the centre which form a large square:

Finally, here is what I think is the most striking of the rep-tile works discussed in this post. There are two different rep-tiles here:

On the upper left is the first rep-tile:

Each rep-tile here is made up of nine squares, and at the same time one can put four of these rep-tiles together to form a square. Therefore, again, this rep-tile can be divided up into thirty-six (four times nine) smaller equally-sized rep-tiles which are replications of the larger rep-tile. We have four rep-tiles on the upper left, two being black and red and two being white and red. This entire square is duplicated in the lower right. The second rep-tile can be seen on the upper right:

Once again this rep-tile is made up of nine squares, and again four of these rep-tiles can be positioned together to form a square. Therefore, again, this second rep-tile can be divided up into thirty-six smaller equally-sized rep-tiles identical in shape to the large rep-tile. On the upper right we see four rep-tiles, two made up of black and blue rep-tiles, and two made up of white and blue rep-tiles. All of this is duplicated on the lower left.

This is the last of the rep-tile posts but a series of posts on extraordinary irrep-tiles and infin-tiles is in the works for some later date.

Next Post: CYCLIC POWER

Post 1 – Infinity Everywhere – https://thekiddca.wordpress.com/2024/04/06/infinities-infinity-everywhere/

Post 2 – Snowflake Curve – https://thekiddca.wordpress.com/2024/04/13/infinities-2-theres-no-business-like-snow-business/

Post 3 – Ch’i Ch’iao T’u – https://thekiddca.wordpress.com/2024/04/20/infinities-3-infinitesimal-chi-chiao-tu/

Post 4 – Hotel Infinity – https://thekiddca.wordpress.com/2024/04/27/infinities-4-no-vacancies-but-rooms-still-available-at-hotel-infinity/

Post 5 – Pythagorean Infinity – https://thekiddca.wordpress.com/2024/05/04/infinities-5-pythagorean-infinity/

Post 6 – Rep-tiles 1 – https://thekiddca.wordpress.com/2024/05/12/infinities-6-welcome-to-the-rep-tile-house-part-1-of-5/

Post 7- Rep-tiles 2 – https://thekiddca.wordpress.com/2024/05/18/infinities-7-welcome-to-the-rep-tile-house-part-2-of-5/

Post 8 – Rep-tiles 3 – https://thekiddca.wordpress.com/2024/05/25/infinities-8-welcome-to-the-rep-tile-house-part-3-of-5/

Post 9 – Rep-tiles 4 – https://thekiddca.wordpress.com/2024/06/01/infinities-9-welcome-to-the-rep-tile-house-part-4-of-5/

INFINITIES 9 – Welcome to the Rep-tile House Part 4 of 5

This is the ninth in a series of posts exploring infinity, entities and ideas infinite, infinities larger than other infinities, infinity paradoxes, infinite perimeters, infinite regress and infinite sequences such as cyclic numbers.

A shape is a rep-tile if it can be divided up into smaller shapes so that all of those smaller shapes are exactly the same shape as the large shape you started with. All of those smaller shapes must also all be the same size for you to have a rep-tile. This is another post about rep-tiles.

This first illustration is pretty straightforward. There are actually two large rep-tiles here, each one a mirror image of the other though they are coloured slightly differently to differentiate them. Each of the two large shapes consists of eighteen squares. However, each of those squares is divided up into two smaller shapes each of which is identical to the large shape. Therefore, each rep-tile consists of thirty-six identical smaller shapes all the same size and all the same shape as the original large shape:

Of course, like all rep-tiles, one can create an infinite regression. In this case each large rep-tile is made up of thirty-six smaller shapes. Each one of those smaller shapes can therefore be divided up into thirty-six shapes which are even smaller and each of those even smaller shapes can be divided up into thirty-six shapes that are yet smaller. One can do that forever infinitesimally. One can of course go in the opposite direction as well, putting thirty-six copies of the large shape you started with together to make an even larger shape, and continue doing that creating ever larger rep-tiles as long as you want.

It occurred to me, however, that one could generate an infinite sequence of rep-tiles in an entirely different way. In an earlier post I discussed a rep-tile that looks like a sort of giant stubby inverted letter T:

One can take four of these inverted T-shapes and put them together to form a square. Since the inverted T-shape is itself constructed out of four squares, one could take each of those four squares and divide each of them up into four smaller T-shapes giving you sixteen T-shapes all together. One could continue creating smaller and smaller (or larger and larger) T-shapes forever.

One could also see this as a very short two-layered staircase, i.e. one could start on the left side, step up onto the first square, take a second step up onto the top square, then step down onto the last square in the staircase, and then take one final step onto the ground again. Then I wondered whether one could create a three-layered reptilian staircase made out of nine smaller squares, with five squares on the bottom layer, three squares as the second layer, and one square at the top. Going from left to right, one could ascend to the top then descend to the ground again on the right side in six steps. Not only that, but one could take four of these three-tiered staircases and form a large square. Since this staircase is itself made out of nine squares, each of those nine squares could be divided up into four smaller three-tiered staircases so that indeed you have a rep-tile which can be divided up into thirty-six (9 x 4) smaller equal-sized copies of itself.

Here is a design I created whose main feature is such a three-tiered staircase:

Besides the three-tiered staircase, this design also features three examples of rep-tiles comprised of two-tiered staircases – Example 1:

Example 2:

Example 3:

Thrown in for good measure is another rep-tile in the shape of an L which I’ve used in earlier posts:

Now comes the interesting part. One could create a four-tiered rep-tile as well, and a five-tiered and six-tiered and so on getting infinitely larger. This isn’t a case of one particular shape increasing in size infinitely and decreasing in size infinitesimally. The three-tiered staircase is not identical in shape to the two-tiered staircase.

Ever larger staircases can be generated so that four of each one can be combined to form a square, and of course each staircase is itself made up of squares. Therefore a staircase rep-tile made up of n squares can be divided up into 4n smaller identical replications of itself. If n=4 you have a two-tiered staircase, and if you put four of them together you have sixteen squares and not only that but those sixteen squares can be used to generate a four by four array consisting of four staircases. Similarly one can construct a six by six array using four three-tiered staircases (each made up of nine squares) since 4 x 9 = 36 = 6 x 6. Here is a diagram illustrating staircases for n = 16, 25, 36, 49 and 64:

NEXT POST: Welcome to the Rep-tile House Part 5 of 5.

Post 1 – Infinity Everywhere – https://thekiddca.wordpress.com/2024/04/06/infinities-infinity-everywhere/

Post 2 – Snowflake Curve – https://thekiddca.wordpress.com/2024/04/13/infinities-2-theres-no-business-like-snow-business/

Post 3 – Ch’i Ch’iao T’u – https://thekiddca.wordpress.com/2024/04/20/infinities-3-infinitesimal-chi-chiao-tu/

Post 4 – Hotel Infinity – https://thekiddca.wordpress.com/2024/04/27/infinities-4-no-vacancies-but-rooms-still-available-at-hotel-infinity/

Post 5 – Pythagorean Infinity – https://thekiddca.wordpress.com/2024/05/04/infinities-5-pythagorean-infinity/

Post 6 – Rep-tiles 1 – https://thekiddca.wordpress.com/2024/05/12/infinities-6-welcome-to-the-rep-tile-house-part-1-of-5/

Post 7 – Rep-tiles 2 – https://thekiddca.wordpress.com/2024/05/18/infinities-7-welcome-to-the-rep-tile-house-part-2-of-5/

Post 8 – Rep-tiles 3 – https://thekiddca.wordpress.com/2024/05/25/infinities-8-welcome-to-the-rep-tile-house-part-3-of-5/

INFINITIES 8 – Welcome to the Rep-tile House Part 3 of 5

This is the eighth in a series of posts exploring infinity, entities and ideas infinite, infinities larger than other infinities, infinity paradoxes, infinite perimeters, infinite regress and infinite sequences such as cyclic numbers.

Illustration copyright MURRAY YOUNG 2024

This is also the third in a series of posts about replicating tiles, or rep-tiles for short. A rep-tile is a shape (e.g. a square) which can be subdivided into smaller shapes all of which are identical in shape to the figure you started with. Those identical smaller shapes are also the same size. If they were different sizes we’d have an irrep-tile (irregular replicating tile). For more details about rep-tiles see the first post in this series.

In my illustration above, there is a pair of rep-tiles in the centre, one on each side of the entrance to the GRÜNER KÄSE MINING CORPORATION. All four of the smaller shapes on each side are the same shape and size. I wonder what GRÜNER KÄSE means? The LUNAR FLYER, however, is made up of two irrep-tiles (irregular replicating tiles).

The top half, for example, is a large shape identical to both the two somewhat smaller brown shapes and the other eight even smaller multi-coloured shapes, but because these ten smaller shapes are not all the same size this is an irrep-tile rather than a rep-tile. In the same way, the two multi-coloured shapes at the bottom are also irrep-tiles rather than rep-tiles:

It is also interesting to note that the two rep-tiles in the centre, the two irrep-tiles at the bottom and the two irrep-tiles that form the Lunar Flyer can all be subdivided into right-angled triangles with angles of 30^o, 60^o and 90^o, three of them, six of them and four of them respectively.

LACUS TEMPORIS
By NASA (image by Lunar Reconnaissance Orbiter) – JMARS, Public Domain, https://commons.wikimedia.org/w/index.php?curid=41050860

Lacus Temporis is an actual location on the moon, but I just made up the name Lunopolis.

The Palace of the Grand Lunar is a structure mentioned in the ground-breaking science fiction novel ‘The First Men in the Moon’ by H.G.Wells published in 1901 and made into films in 1919 and 1964. One of the two protagonists in that novel is a scientist named Mr. Cavor.

Most of the shapes in the illustration above are rep-tiles. In the left and right corners at the top, however, are two irrep-tiles with smaller and smaller squares eventually becoming infinitesimal:

In this illustration we see two colourful red, orange and green trapezoids at the top which are also rep-tiles, each divided into nine smaller trapezoids:

A trapezoid is any four-sided figure with one pair of parallel sides. In between these two trapezoids it can be seen how one can put three trapezoids together to form an equilateral triangle. Underneath that triangle is another trapezoid made up of smaller and smaller trapezoids which makes this version of the trapezoid an irrep-tile. At the bottom are all manner of some of the less interesting rep-tiles in this world, including:

a rhombus on the far left (four equal sides)
two varieties of right-angled triangles with angles of 30^o, 60^o and 90^o
isosceles right-angled triangles
parallelograms (a quadrilateral with opposite sides parallel)
symmetrical trapezoids
squares
asymmetrical trapezoids
irregular pentagons
an equilateral triangle
a rectangle

Six of the shapes here are rep-tiles. The three similar rep-tiles here in blue, brown and black, are different from but similar to the six large rep-tiles in the first illustration at the beginning of the first post in this series.

Also, apart from those six rep-tiles there is an irrep-tile in the bottom left hand corner:

NEXT POST: Welcome To The Rep-Tile House Part 4

INFINITIES 7 – Welcome to the Rep-tile House Part 2 of 5

This is the seventh in a series of posts exploring infinity, entities and ideas infinite, infinities larger than other infinities, infinity paradoxes, infinite perimeters, infinite regress and infinite sequences such as cyclic numbers.

Illustration copyright M.W.F. YOUNG 2024

This is also the second in a series of posts about replicating tiles, or rep-tiles for short. Some rep-tiles are quite boring but some are very interesting. A rep-tile is a shape (e.g. a square) which can be divided into smaller shapes all of which are identical in shape to the figure you started with. Those identical smaller shapes are also all the same size. For more details see the first post in this series on rep-tiles.

There are seven rep-tiles in the illustration above. For example, take a look at the figure on the left at the top.

The overall large figure consists of nine squares, each of those squares consists of four smaller shapes (blue, orange, grey and brown), and each of those smaller shapes is identical in shape to the large overall shape you started with. You have thirty-six (9 x 4) of the smaller shapes inside the large shape. Not only that, but all thirty-six smaller shapes are all the same size. This large shape is a rep-tile.

THE TRIAL OF GIORDANO BRUNO
By Jastrow – Self-photographed, Public Domain, https://commons.wikimedia.org/w/index.php?curid=1193142

All seven rep-tiles here consist of nine squares and each square is divided up into four smaller shapes. The text consists of historical dates and events that happened during each year listed. For example, the Roman emperor Marcus Aurelius was born in 121 CE. In 1600 Dominican friar and later cosmologist Giordano Bruno is burned at the stake for heresy. He had insisted that the universe was infinite and therefore could have no centre. He also rejected the afterlife, the Trinity and other central church doctrines. The entry under the date 2025 is speculative.

With these rep-tiles we have squares made up of small rep-tiles and large rep-tiles made up of squares – squareness is the name of the game. Have you guessed the significance of all these dates / numbers? The dates / numbers are themselves perfect squares: 121 = 11 x 11. 1600 = 40 x 40. 1936 = 44 x 44 and so on. You might want to test your number sense by going through and trying to guess what the square root of each date / number is.

Did you also notice the four numbers at the bottom without text – 5, 25, 625, 390625? It seems that if you square 5 you get 25 and the last digit of 25 is 5. If you square 25 you get 625 and the last two digits of 625 are 25. If you square 625 you get 390625, and the last three digits of 390625 are 625.

SPHINX

There is also something called an IRREP-TILE which is an IRREGULAR REPLICATING TILE. An irrep-tile, like a rep-tile, is a shape which can be subdivided into smaller shapes all of which are identical to the large shape you started with. However, those smaller shapes are not all the same size. In the illustration above, featuring a shape which mathematicians refer to as The Sphinx, you can see four rep-tiles at the bottom which are four different ways in which the sphinx shape can be subdivided into nine identical smaller sphinxes.

However, every rep-tile, not just this one, can be turned into an irrep-tile, as illustrated by the main figure in this illustration. In this main figure one can see that the Sphinx has been subdivided into four smaller sphinxes, separated here by thin white lines. However, any one of those four (in this case the darkest one) can be subdivided into four smaller sphinxes, and any or all of those four smaller ones can be subdivided further, and so on forever until the shapes are infinitesimally small.

This is my illustration of another irrep-tile. I started with a configuration of four squares in the shape of a sort of stubby upside-down T shape. Each one of those four squares can be subdivided into four smaller T-shapes identical to the original large T-shape. The T-shape, treated as a rep-tile, can therefore be subdivided into sixteen smaller shapes identical to the original and all the same size. I’ve subdivided just the three bottom squares each into four smaller T-shapes, each one in the form of part of a maze:

By the way this maze is a legitimate maze, with an entrance and an exit and a way to get from one to the other. I also subdivided the fourth square, at the top, into four smaller T-shapes (divided by thin white lines):

However in this case I subdivided two of those four T-shapes (at the bottom and on the right) into sixteen smaller T-shapes – in the bottom case two dark blue, two light blue, six brown and six orange T-shapes, and in the case on the right six dark blue, six light blue, two brown and two orange T-shapes.

In the case of the last two T-shapes (on the left and at the top) in the square at the top, in each case I subdivided the T-shape into fourteen smaller T-shapes plus two other T-shapes each of which has been subdivided further into sixteen white T-shapes.

I could have kept subdividing forever until the shapes were infinitesimally small. On the other hand I could have just divided the original large inverted T-shape into sixteen smaller T-shapes in the form of one large maze.

NEXT POST: Welcome To The Rep-Tile House Part 3

INFINITIES 6 – Welcome to the Rep-tile House Part 1 of 5

This is the sixth in a series of posts exploring infinity, entities and ideas infinite, infinities larger than other infinities, infinity paradoxes, infinite perimeters, infinite regress and infinite sequences such as cyclic numbers.

The term rep-tile was coined by Solomon W. Golomb, an abbreviated word based on the phrase REPLICATING TILE. The simplest rep-tile is a square. If you take a square and join the mid-points of opposite sides you end up with four smaller shapes all the same shape as the large square you started with, and they are all also the same size. Therefore the square is a rep-tile. The same thing works with any rectangle, parallelogram, rhombus and equilateral triangle. Boring. However in the illustration at the top of this post is a set of six rep-tiles which are all a lot more interesting than a square. In each case the large shape can be divided up into smaller shapes all of which are the same shape as the shape you started with, and all of those smaller identical shapes are also the same size.

In every illustration I have inserted a reptile, in this case a gecko. The bee on the left is called Slim Harpo – it was Slim Harpo (real name: James Moore) who composed and recorded the song called ‘I’m a King Bee’, later covered by The Rolling Stones, among others.

Also, the bee on the right has a tag attached to it reading “IF FOUND PLEASE RETURN TO SHERLOCK HOLMES”. When Sir Arthur Conan Doyle had his creation, Sherlock Holmes, retire to the Sussex Downs in his later years, Doyle tells us that one of the things that fascinated him in his retirement was his study of the segregation of the queen in bee hives.

Then there’s the queen bee labelled Queen Elizabeth.

THE REAL QUEEN ELIZABETH II.
By Ministry of Information official photographer – http://media.iwm.org.uk/iwm/mediaLib//20/media-20543/large.jpgThis photograph TR 2832 comes from the collections of the Imperial War Museums., Public Domain, https://commons.wikimedia.org/w/index.php?curid=24396386

ISOSCELES RIGHT-ANGLED TRIANGLES

In this second illustration, there are seven rep-tiles all constructed out of isosceles right-angled triangles. For example, take the shape at the top on the left.

It consists of four connected isosceles right-angled triangles each of which is subdivided into smaller blue and yellow isosceles right-angled triangles. If you examine any of these eight configurations of smaller triangles (four blue and four yellow) you’ll discover that each is identical in shape to the large shape you started with originally.

This figure in the middle on the right, with the smaller triangles, is particularly ingenious. The rep-tile just below that one is also interesting in that each of the eight smaller sets of yellow and blue triangles (two in each set) can also be viewed as four sets of yellow and blue triangles (four in each set) each of which consists of rep-tiles identical to the large shape on the left at the top of the main illustration (the first rep-tile discussed above):

The preponderance of RIGHT-angled triangles led to the idea of adding phrases which include the word RIGHT. Do The Right Thing is a landmark 1989 feature film produced, written and directed by Spike Lee. Right Now is a wonderful sort of ingenious meta-music video released by Van Halen in 1991 – https://www.youtube.com/watch?v=gU7d2EHV_OQ&ab_channel=VanHalen . I’m All Right, Jack is a 1959 British film starring Peter Sellers who won a BAFTA for his performance in the film. The Right Stuff was a 1983 film about the test pilots selected to be astronauts, all of them male, as part of the misogynist early American space program (though the film seems to have overlooked the misogyny). For more details see my previous post here – https://thekiddca.wordpress.com/2021/07/31/negative-space/ .

SQUARES

In this final illustration, everything is based on squares. On the left at the top the large shape consists of eight smaller squares, and each smaller square is divided up into an orange shape and a green shape, making sixteen smaller shapes in all. Each of those sixteen smaller shapes is identical to the large shape you started with.

The same thing applies to the other shapes across the top. The trick is figuring out how to divide a square into two identical halves each of which consists of eight small squares.

On the second row we have fourteen large L-shaped rep-tiles each of which is sub-divided into nine smaller L-shapes, which can be done in fourteen different ways as illustrated here.

Each large L-shape could also be sub-divided into four identical squares. The illustration also shows that one can put four large L-shapes together to create a large square, like the three pictured here.

NATHAN PHILLIPS SQUARE, TORONTO.
By Hutima – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=121721123

The rest of the illustration runs with the concept of squareness, e.g. the sailing ship is a square rigger.

In the vicinity of that ship are various square-related entities from the field of Mathematics. At the bottom are the names of various famous squares:

Trafalgar Square in London, England – with Nelson’s Column surrounded by lions
St. Peter’s Square in Vatican City, Italy – fronting St. Peter’s Basilica, home of the Pieta
Tiananmen Square in Beijing, China – where Mao Zedong proclaimed the birth of modern China in 1949, and many died in 1989
Leicester Square in London, England – referenced in songs, including Jeffrey Goes to Leicester Square (Jethro Tull), It’s a Long Way to Tipperary (from World War One), and a rather graphic song by The Rolling Stones
St. Mark’s Square in Venice, Italy – which Napoleon called the drawing room of Europe
Nathan Philips Square in Toronto, Canada – site of many an anti-war demonstration
Wenceslas Square in Prague, The Czech Republic – site of demonstrations of more than a hundred thousand people during the Velvet Revolution
Red Square in Moscow, Russia – site of Lenin’s Tomb and St. Basil’s Cathedral

ONE OF THE LIONS AT THE BASE OF NELSON’S COLUMN IN TRAFALGAR SQUARE, WITH BIG BEN IN THE BACKGROUND
By Florinux – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=12463608

There’s a square knot in the bottom right hand corner, and a variety of mathematical and linguistic squares. In the four magic squares on the left all numbers in all of the rows, columns and diagonals add up to the same number.

These squares exhibit other interesting mathematical properties as well, e.g. in the magic square at the top on the right, if you reverse the digits in all 16 numbers (96 becomes 69 etc.) you still end up with a magic square.

In the set of four squares on the right, each row, column and diagonal of the top left square, a Latin square, contain all four digits – 1, 2, 3 and 4.

In the word square on the right at the top, the four words reading across (card, area, rear, dark) are repeated reading down. In the bottom left word square there are four words reading across, but four different English words reading down. In the last word square there are four words reading across (tans, area, lion, land), four different words reading down (tall, aria, neon, sand), and two words reading diagonally (trod, lies).

NEXT POST: Welcome To The Rep-Tile House Part 2

Post 1 – https://thekiddca.wordpress.com/2024/04/06/infinities-infinity-everywhere/

Post 2 – https://thekiddca.wordpress.com/2024/04/13/infinities-2-theres-no-business-like-snow-business/

Post 3 – https://thekiddca.wordpress.com/2024/04/20/infinities-3-infinitesimal-chi-chiao-tu/

Post 4 – https://thekiddca.wordpress.com/2024/04/27/infinities-4-no-vacancies-but-rooms-still-available-at-hotel-infinity/

Post 5 – https://thekiddca.wordpress.com/2024/05/04/infinities-5-pythagorean-infinity/

INFINITIES 5 – Pythagorean Infinity

This is the fifth in a series of posts exploring infinity. There will be one infinity larger than another infinity, infinite perimeters, infinite regress, infinity paradoxes, an infinite set of bizarre things called cyclic numbers, and other infinity-related concepts.

Illustration by M.W.F. YOUNG based primarily on a diagram by John Waterhouse published in July 1899, reprinted in *The Pythagorean Proposition* by Elias Loomis.

Pythagoras (570 BCE – 490 BCE) is famous for his theorem which states that the square of the hypotenuse is equal to the sum of the squares on the other two sides. In the illustration above, one can see a right-angled triangle with three squares on its sides as part of an ever larger design with ever larger right-angled triangles. One could keep enlarging this forever and get an infinitely large design. One could also go in the opposite direction toward the infinitesimal. The design is also pictured in negative form next to it, and around the outside are graphic proofs of the Pythagorean Theorem, just a few of the hundreds of different known proofs of the theorem.

It should be noted that Pythagoras never claimed to have come up with the first proof of his famous theorem, and in fact the theorem had indeed been proved long before Pythagoras was born. A form of the theorem was known in China about a thousand years before the time of Pythagoras (and Pythagoras was born about 2600 years ago). It also happens to be the case that the area of a regular pentagon constructed on the hypotenuse of a right-angled triangle is equal to the sum of the areas of regular pentagons constructed on the other two sides of that triangle. The same is true of semicircles constructed on the sides of a right-angled triangle.

BERTRAND RUSSELL 1949
By Yousuf Karsh for Anefo – Nationaal Archief: entry a96fca82-d0b4-102d-bcf8-003048976d84, CC0, https://commons.wikimedia.org/w/index.php?curid=75100590

Pythagoras also devised and analysed the system we use to construct musical intervals. If you like music, thank Pythagoras. This was the first example of a description of a physical phenomenon mathematically so one could argue that Pythagoras invented Mathematical Physics. As a philosopher his ideas influenced philosophical giants such as Plato and Aristotle who came later. His ideas also influenced legendary scientists centuries later such as Copernicus, Kepler and Newton. In the twentieth century the great logician, mathematician and philosopher Bertrand Russell was an enthusiastic Pythagoras fanboy. Pythagoras came up with the Theory of Proportions, he was quite aware that the Earth was spherical, and he deduced that both the Evening Star and the Morning Star were both the same celestial object, i.e. the planet Venus. Despite his fame and influence, he lived so long ago that we don’t even know for sure when or where he died.

PYTHAGORAS TEACHIN BOTH WOMEN AND MEN
FROM A 1913 PAINTING
By Internet Archive Book Images – https://www.flickr.com/photos/internetarchivebookimages/14783288925/Source book page: https://archive.org/stream/storyofgreatestn01elli/storyofgreatestn01elli#page/n586/mode/1up, Public Domain, https://commons.wikimedia.org/w/index.php?curid=42033593

There are some nations today who refuse to allow women to get an education. When activist teenager Malala Yousafzai attempted to do something about that the Taliban shot her point blank in the head. She survived and went on to become the youngest recipient of the Nobel Peace Prize. About 2570 years ago Pythagoras of Samos formed a secret group of followers enthusiastically studying Mathematics and he welcomed women to join. 2570 years ago Pythagoras was more civilized than the Taliban today. We know of a least seventeen women who joined with Pythagoras.

MALALA YOUSAFZAI
By
内閣官房内閣広報室
– kantei.go.jp –
ユスフザイ女史による表敬及び共同記者発表
, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=104706553

Here is another rendering of the figure in the illustration at the top of this post. I based this is illustration on a diagram by John Waterhouse published in July 1899, reprinted in The Pythagorean Proposition by Elias Loomis.:

PREVIOUS POSTS –

INFINITIES 4 – No Vacancies But Rooms Still Available At Hotel Infinity

This is the fourth in a series of posts exploring infinity. There will be one infinity larger than another infinity, infinite perimeters, infinite regress, infinity paradoxes, an infinite set of bizarre things called cyclic numbers, and other infinity-related concepts.

A lot has been written about Hotel Infinity. Hopefully you will learn something new here. Here’s the set-up. Imagine a hotel containing fifty rooms, and there is a single guest in each room. If a new guest comes along and wishes a room she will be out of luck. Now imagine a hotel with an infinite number of rooms, with a guest in each room. Lo and behold a new guest can still be accommodated while still giving every guest their own room. It was German logician David Hilbert who came up with the idea of Hotel Infinity as a way of demonstrating the surprising and counter-intuitive properties of infinite sets (one of the people Hilbert worked with at Göttingen University in 1915 was a young man you may have heard of named Albert Einstein).

DAVID HILBERT
By Unknown author, derivative work Lämpel – Reid, Constance (1970) Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 230 ISBN: 978-3-662-27132-2., Public Domain, https://commons.wikimedia.org/w/index.php?curid=130481740

Infinities can be quite perplexing. For example, the part may be the same size as the whole! This is demonstrated by the fact that there are just as many natural numbers(1, 2, 3 . . . ) as there are even numbers (2, 4, 6 . . .) because you can match 1 up with 2, 2 up with 4, 3 up with 6 and so on. The idea of Hotel Infinity was first introduced by Hilbert in his 1924 lecture “Über das Unendliche” (“On The Infinite”), and Hilbert’s idea was first popularized by George Gamow in his excellent and quite accessible book ‘One, Two, Three . . . Infinity’ published in 1947 (page 17). The celebrated Polish science fiction writer Stanislaw Lem (1921 – 2006) also wrote a story called ‘The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet’. It was first published in 1968 as part of the anthology ‘Stories About Sets’ edited by N. Ya. Vilenkin. It takes place at the Hotel Cosmos which has an infinite number of rooms.

STANISLAW LEM IN 1966
By Courtesy of Lem’s secretary, Wojciech Zemek. Resize and digital processing by Masur. – Stanislaw Lem 2.jpg, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=915059

Here’s how the apparently impossible can be achieved. Hotel Infinity is a hotel with an infinite number of rooms, each room is occupied by a single guest, and none of the guests is willing to share their room with anyone else. A new guest arrives asking for a room for the night and the Hotel Manager is happy to oblige. In order to accommodate the new guest, the Hotel Manager can simply move Guest 1 from Room 1 into Room 2, move Guest 2 from Room 2 into Room 3 and so on. Then place the new guest into Room 1. In Hotel Infinity there will always be new rooms to move guests into.

Similarly, one can accommodate, say, five new guests. Move Guest 1 from Room 1 into Room 6 (1 plus 5), move Guest 2 from Room 2 into Room 7 (2 plus 5) and so on. Therefore one can accommodate g new guests by moving Guest 1 into Room 1 + g, move Guest 2 into Room 2 + g and so on. The Hotel Manager can, therefore, accommodate any finite number of new guests since g can be as large as you like.

What is more amazing is that one can also accommodate an infinite number of new guests. Simply move Guest 1 from Room 1 into Room 2, move Guest 2 from Room 2 into Room 4, move Guest 3 from Room 3 into Room 6 and so on. All the guests are now in the even numbered rooms and all the odd numbered rooms are empty. Since there are an infinite number of odd numbers, one can accommodate an infinite number of new guests.

What about an infinite number of infinite sets of new guests? Easy. First of all, do what you did in the previous case and free up all the odd numbered rooms again. Now take a look at the array in my illustration at the beginning of this post depicting an infinite number of guests in each of Bus 1, Bus 2, Bus 3 and so on, in fact an infinite number of buses. By following the arrows you can see that one can select all of the passengers on all of the buses one at a time and place them in those odd numbered rooms without overlooking any of the bus passengers. These infinities are called countable or denumerable infinities.

CANTOR’S DIAGONAL DIAGRAM IN HIS ARGUMENT FOR THE EXISTENCE OF UNCOUNTABLE SETS
By Cronholm144 – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2206106

THE MOST EXTRAORDINARY THING OF ALL

Despite all these ingenious methods of accommodating larger and larger sets of infinities in Hotel Infinity, sometimes you can’t accommodate everyone new. The infinities we’ve been talking about are countable infinities. However it has been proven with mathematical rigour (over a century ago, by Georg Cantor), that some infinities are uncountable, and are larger than countable infinities, as bizarre as that sounds. The number of Natural Numbers (one, two, three etc.), designated aleph null, is countably infinite. However the number of Real Numbers, designated aleph one, is not countable, so there are more Real Numbers than there are Natural Numbers even though the Natural Numbers are infinite. If the size of the set of new guests was the same size as the size of the Real Number set then some of the guests would be out of luck.

NEXT POST – Pythagorean Infinity

POSTSCRIPT: More Advanced Accommodation Strategies:

There is a second way of accommodating an infinite number of infinite sets – The Prime Powers Method. Start with powers of two. Before the buses arrive rearrange the occupants who are already in the hotel. Put the guest from Room 1 into the room numbered two to the power 1 (Room 2). Put the guest from Room 2 into the room numbered two to the power 2 (Room 4). Put the guest from Room 3 into the room numbered two to the power 3 (Room 8). Continue so that all infinite guests are now in the infinite rooms numbered with the infinite powers of two (two is the first prime number). When Bus 1 arrives place its infinite occupants into the rooms numbered with the infinite powers of three (three is the second prime number). Place the infinite occupants of Bus 2 into the rooms numbered with the infinite powers of five (the third prime), and place the infinite occupants of Bus 3 into the rooms numbered with the infinite powers of seven (the fourth prime). Continue doing this forever, for as Euclid first proved about 2300 years ago the number of primes is infinite. It is because we’re using primes that none of these infinite prime power sets overlap. Furthermore, this way you also end up with an infinite number of unoccupied rooms.

We used the Prime Powers Method to accommodate two layers of infinity (an infinite number of infinite sets). There are at least four other different ways to accommodate these two layers of infinity: The Prime Factorizations Method, The Interleaving Method, The Triangular Number Method and The Arbitrary Enumeration Method. The nice thing is that one can also use those four methods to accommodate three, four, five, and in fact an infinite number of levels of infinity.

The Prime Factorizations Method, for example, involves infinite powers of prime numbers. All you need to do is put a new guest who was sitting in seat s of bus b into the room designated by the product of 2 (the first prime) to the power s and 3 (the second prime) to the power b. For example a new guest sitting in seat 4 of bus 5 would go into room 3888 (16 x 243). The value of b would be zero for the people already in the hotel. Since every number has a unique prime factorization, this solution even leaves some rooms empty. Here is a link to more details – https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel .

INFINITIES 3 – Infinitesimal Ch’i Ch’iao T’u

This is the third of a series of posts exploring infinity. There will be one infinity larger than another infinity, infinite perimeters, infinite regress, infinity paradoxes, an infinite set of bizarre things called cyclic numbers, and other infinity-related concepts.

Ch’i Ch’iao T’u is the Chinese name for a tangram set which consists of seven shapes called tans which can be used to create all manner of ingenious shapes. The seven pieces consist of two large triangles (each with an area of four square units), a medium sized triangle and a parallelogram and a square (each with an area of two square units) and two small triangles (each with an area of one square unit). One ingenious thing about these seven shapes is that you can put them together to form a giant triangle, a giant square and a giant parallelogram (as well as a giant trapezoid, and a giant rectangle). Some people use tangram sets to teach geometry. In the illustration at the top of this post the shapes are used to form a giant right-angled triangle consisting of the five triangles, square and parallelogram that each tangram set contains. In that illustration each of those seven shapes are then subdivided again into smaller replicas of those same seven tans that we started with. One can keep doing this forever creating smaller and smaller tans infinitesimally, as illustrated. Here is a second illustration of mine demonstrating that same idea of infinite regression:

Tangrams were first invented some time during the reign of Chia-ch’ing (1796 – 1820) in China. The first tangram set was invented by someone using the manufactured name Yang-ch-chu-shih which in English means dim-witted recluse. The corresponding English word ‘tangram’ was invented by Thomas Hill in 1848. Tans can also be used to create the silhouettes of people, animals, plants, buildings, letters, numbers and miscellaneous objects. There have been periodic crazes in which people are challenged to make these shapes, and many others, with tangram sets. I have a book in my mathematical library with 1756 different shapes that can be made using a tangram set. Here is my illustration of a giant set of tans in the form of a square (surrounded by other shapes), the entire design consisting of 2318 tiny tans:

In the illustraion at the top of this post, on the left side, there is a geometric proof of the Pythagorean Theorem for isosceles right-angled triangles, illustrated using tans. Here is a fourth illustration of mine, this one showing a similar proof:

In 1942 Fu Traing Wang and Chuan-Chih Hsiung proved that there are only 13 convex tangram configurations. I used that idea to construct a giant rectangle using only those 13 configurations. Notice that the configuration pattern in the bottom half of this rectangle is a mirror image of the pattern of configurations in the top half:

Here is a detail from the above illustration showing how each configuration is constructed from a tan set:

Over the centuries people have constructed tangram sets out of wood, mother of pearl, precious metals and ivory. Some sets are finely carved works of art. Dozens of books have been published about tangrams, many with shapes illustrated with comic figures or detailed scenes. There are dishes in the shapes of tans, and tangram table sets which can be rearranged to accommodate different numbers of people. People have written stories revolving around tans, and Edgar Allen Poe owned a set carved from ivory. Napoleon may also have been a tangram puzzle lover. Much has been written about tangrams but don’t believe anything written by the famous American puzzle master Sam Loyd. He wrote a book about the history of tangrams but he made everything up out of whole cloth. Here is a final illustration of mine highlighting tans:

NEXT POST: No vacancies but rooms still available at Hotel Infinity.