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 5 – https://thekiddca.wordpress.com/2024/05/04/infinities-5-pythagorean-infinity/
According to The Kidd 