May 2024 – According to The Kidd

PLUS ÇA CHANGE, PLUS C’EST LA MȆME CHOSE

I saw a news item today on CNN. Someone named Ed Dwight is about to travel into space on the new Shepherd rocket sent up by Blue Origin. Dwight is African-American. John F. Kennedy selected him personally for astronaut training back in 1961 and if he had made it into space he would have been the first African-American to do so. He never made the cut back then but he is going to go into space now, finally. He is ninety years old. More power to him. If racism was part of the reason why he wasn’t chosen to go into space in the 1960’s then it’s good to see that things seem to be better now, at least as far as race is concerned.

This news story made me think of the return to space of American astronaut John Glenn as a senior citizen. The Soviets took an early lead in the Space Race back in the 1960’s. They took the first photographs of the dark side of the moon (which never faces Earth), they put the first satellite into space (Sputnik), the first living creature into space (the dog Laika), and the first man (Yuri Gagarin) all before the Americans finally put their first space traveller, John Glenn, into space in 1961. Like Ed Dwight, Glenn became a senior citizen in space, when he returned to space in 1998 at the age of 77.

First of all, President Kennedy may have selected Ed Dwight personally, but Kennedy also specifically made sure that there were no women in the U.S. Space Program. His successor, President Johnson, was also adamantly opposed to allowing women into the space program. The Soviets put the first woman into space (Valentina Tereshkova), and later they put a second woman into space, Svetlana Savitskaya, while the U.S. were still banning women from their space program. In 1998 the U.S. had the chance to make amends for their misogyny by choosing a woman to go up but they chose the 77 year old Glenn instead.

VALENTINA TERESHKOVA 1969
By RIA Novosti archive, image #612748 / Alexander Mokletsov / CC-BY-SA 3.0, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=18560617

It seems that when the choice is between a man or a woman, equally suited for the job, the powers that be in the U.S. make sure that the man wins. The U.S. is known for its racism but in 2008 when the choice was between the first African-American president (Barack Obama) and the first female president (Hilary Clinton), it was Obama who became president. Remember too, in 2016, when the choice was between an unqualified white male who bragged about being a sexual predator and had a history of overt racism (Donald Trump) and a better qualified woman (Hilary Clinton), it was Trump who won.

You might say that maybe women were barred from the U.S. Space Program back in the 1960’s because men and women were not equally qualified. You would be right. But the women were MORE qualified than men. This is explained in the following excerpt from a previous post I did on this matter:

“In 1959 a space researcher named William Randolph Lovelace began investigating the idea of sending women into space because there had been scientific studies in the United States, the United Kingdom and Canada showing that women were more capable than men at enduring long periods of time in cramped space, and enduring sustained isolation. There were data showing that women sustained fewer heart attacks than men, and women had an internal reproductive system which was more protected from radiation, vibration and violent shaking than was the case with men. Also, women being on average lighter than men was an advantage as that meant fewer supplies to sustain a woman, and less fuel to carry her.

William Lovelace, and Brigadier General Donald Flickinger put together a testing regimen which was similar to and AT LEAST AS DIFFICULT to pass as that given to candidates for the Mercury Project which trained male astronauts. The testing was rigorous and draining, at times painful and invasive, and took several tightly-scheduled days to complete. They had to supply incredibly detailed family and medical histories, and they were checked exhaustively for any physical imperfections such as minor heart problems which may be inconsequential on the ground but may become problematic in space. There were also psychological tests involving isolation periods and sensory deprivation.

When John Glenn was asked about whether women should be allowed to be part of the U.S. Space program, this was what he said:

“It is just a fact. The men go off and fight the wars and fly the airplanes and come back and help design and build and test them. The fact that women are not in this field is a fact of our social order.”

When Valentina Tereshkova went up I was still a young lad, and up here in Canada my friends and my teachers and I realized that her flight was a big deal worthy of praise even though she was from a Communist state. A major scientific achievement isn’t any less of a scientific achievement if the person who did the achieving doesn’t share your particular political views. Anyway, at the time I didn’t realize that Russia was (and is) communist in name only. But the establishment in the United States mocked and trivialized Tereshkova’s achievement, just as the male American astronauts savagely made fun of the idea of a female astronaut. It pissed me off then and it pisses me off now – hence this post.

Here is my earlier complete post on the matter – https://thekiddca.wordpress.com/2021/07/31/negative-space/

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

Illustration copyright MURRAY YOUNG 2024

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.

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

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

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.

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.:

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