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.
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 30o, 60o and 90o, 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 30o, 60o and 90o
- 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
- 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/
According to The Kidd 
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