Technically, it is not infinity. But practically, it is.
You would waste your whole life if you tried to count all the grains of sand on the nearest beach.
You would return to dust before you can count every individual cell in your body. Not only because you are continuously producing new ones, or because you can't see any of them.
In that sense, you are touching infinity, everywhere around and inside; you are made of "infinity".
So... it shouldn't be that astonishing to also find this kind of "infinity" in one single row of the Fractal Yupana.
Multiply the age of the universe by itself, and multiply that result by the age of the universe... and do that again with the lastest result... 10 times ! and then multiply that by 10 billions...
The result, in years, approximately 10 billions Googols (https://en.wikipedia.org/wiki/Googol) years, is the amount of time required to count from 1 to 100, by all the different ways there are to do that, by using a single row of a Fractal Yupana.
(This, assuming a few things, including the speed of your manipulations on a real yupana, taking no rest at all, living 1 Googol times as long as the current age of the universe... at least the universe is set to exist long after that https://www.science.org/content/article/way-universe-ends-not-whimper-bang )
By comparison, there is only 1 way to count from 1 to 100 with a Soroban. So, that would not be a way to appreciate "infinity".
Illustrations for an apparently unsubstantiated claim.
Number of ways you can "write" each number using a single row of the Fractal Yupana.
You can represent every number from 0 and 100 on one single row of a Fractal Yupana. Even though you would not do that, normally, except during multiplications, prefering to represent numbers from 0 to 9 on each row.
3 followed by 115 zeros... 3 quadrillons of googols !?
That is less than the total number of the different ways to count from 0 to 100 on a single row of a Fractal Yupana.
It is the result of the multiplication of the number of representations of each of the numbers from 0 to 100. That is a multiplication of 101 terms, and it is going to be huge (even though the 3 first ones and the 3 last ones are equal to 1). For instance, you have already 20 numbers that have at least 35 different representations: those from 38 to 57. That only is already greater than 35 multiplied by itself 20 times.
Comparison with other interpretations of the Yupana.
Let's first show the distribution of the numbers of ways to represent each number on one single row of the yupana. Then we'll compare the total number of ways to count from 0 to the maximum value on 1 row.
Yupana Dinamica / Inka.
You can count from 0 to 39 on a single row of the Yupana Dinamica (assuming, as for the Fractal Yupana, that you put 0 or 1 token on each spot). For instance, 145 is the number of ways to represent 18, same thing for 21.
This may seem very grossly exaggerated. It is due to the fact that we count as being different two different spots in the same column, even though all spots in one column have the same value (example, each of the 5 spots of the 4th column have value 5). But we need (? not really) to do that if we are to compare with the interpretations of the Yupanas that give a different value to each spot.
If we consider as identical all the spots of one column, the number of ways to write a number falls down drastically. For instance, you'll have only 3 ways (5, 3+2, 2+2+1), instead of 12, to represent 5. And instead of 49 ways to represent 10, you'll have only 3 (5+5, 5+3+2, 5+2+2+1).
Yupana Chirinos-Rivera.
You can count from 0 to 66 on a single row of the Chirinos-Rivera's Yupana (assuming, as always, that you put 0 or 1 token on each spot). For instance, 70 is the number of ways you can represent 33.
Some civilizations are said to not have names for quantities/numbers greater than one and two.
You can find interesting considerations about that on "The Number Warrior" blog article: Is one, two, many a myth (see link at the end of this article if you are interested).
note, 2024/3/3 : i just read some articles about "toki pona", an interesting conlang containing (originally) only 120 words, and only 3 words for numbers (wan=1, tu=2, mute=many); and about some efforts by the speakers to try to complete that set with other names, even trying to promote non-base 10 systems in order to keep close to the original set of 3 names for numbers. (If interested, try to read from here: https://sona.pona.la/wiki/Recommended_learning_resources).
Not having names for numbers greater than two doesn't prevent you from having an accurate representation for quantities. For example, you can still express exact quantities like "two and one", "two for each of your fingers", "two groups of two for each of the joints of your body", "two groups of two and one group of two and one", "two and one for each of the fingers of the members of our tribe" (and that's an accurate quantity, at least for an instant, even if you don't know how many members are in your tribe, as long as none of them looses a finger before you finish your sentence).
And not having names for numbers doesn't even prevent you from performing calculations. We are going to show that, with the example of multiplications performed with the Yupana, the (mythical or real) counting board of the Inca civilization. (Ok, the Inca civilization had names for numbers, but it's irrelevant here... as we plan to badly exaggerate with this concept).
note, 2024/3/3 : the previous remarks would not be totally true for "toki pona": it has no word for "fingers", "joints"... So... you can still have accurate mental representations for numbers by mentally using those equivalences; you would just find it hard to express them verbally, but that would not prevent you from expressing them graphically, or physically (with a counting board, or a khipu, or gestures, etc.).
First, there have been various hypotheses and representations for the Yupana and for the way it was used. But the most general representation is a plane counting board with rows and cells permitting to describe numbers in base ten. (All of that resembling a lot to the north European counting boards still in use in the XVII th century... hmm, does that seem suspect ? or is it just obvious ? because there may not be a lot of ways to design and use a counting board with numbers expressed in base 10).
Modern Yupana
Example below: image of a Yupana promoted under the name "Yupana Inka". It is generally used with a common method where the position of a token in one cell has no particular meaning, the number of dots/spots in a cell representing the value of the each of the tokens that will be put in the cell... knowing that each horizontal row represents a power of 10.
(image extracted from researgate.net - see link to the original at the end of this article)
Let's try to use only "one, two" to describe the standard Yupana, as "one, two" is enough to describe any size of "many" (ok, "one" alone is already enough for that, but well, it's easier with "one, two") :
- in each row, cells with "one", "two", "one and two", "one and a pair of twos" dots/spots.
- each token that will be put on the board will have the value of the cell in which it is put.
- value of one cell in the first (bottom one) row = quantity of dots/spots in it.
- from the second row upward, value of one cell = as many "pairs of the quantity represented in the leftmost cell of the row underneath" as there are dots/spots in the cell... or ..."twice the value of the leftmost cell in the row underneath" for each dot/spot in the cell... or accumulation, for "each finger in a primate" of the "value of the cell underneath";
- a column can also be attributed a value: number of dots/spots in any one of its cells
(ok, yes, we could orientate the counting board in various ways without really changing its interpretation and workings, we would just need to modify the spatial words in the description).
note, 2024/3/3 : and, ok, using "toki pona", we would not easily be able to formulate verbally those explanations... due to the lack of words, we would probably need a bunch of the existing "toki pona" words, with their caracteristically hyper-broad meaning, to try to describe a "row", a "column", a "spot", a "dot", etc... I don't think i'm going to try that, even if i learn a bit more of "toki pona" :-) . The other solution would be, as for every specialized human activity (think medecine, computers, linguistic, philosophy, etc.) to create a specially adapted set of technical words, designed to describe the matter, and only understood and used by the specialists.
Historical Yupana
You can see here a copy of the mythical/historical only known drawing of a Yupana, represented near an Inca accountant holding a quipu (a link to the original image is present at the end of this article).
We can notice that for some cells, some of the dots in the cell are colored differently, which may represent the presence of a token at a particular spot in the cell, with a particular value depending on this spot in this cell.
Following this principle, we can use a "fractal" interpretation of the Yupana that will allow big multiplications (and, obviously, also additions and subtractions, but that is already easy with the common use of the Yupana for which, instead, big multiplications may be cumbersome, unless we use a diversity of tokens having different values: one, two, three, five, for example). Only difference with the standard interpretation of the Yupana: the value of a token in a cell depends on its position in the cell.
Fractal Yupana
In this fractal representation of a Yupana, we have horizontal rows of rectangular cells that contain elliptic spots that contain rounder dots (that contain nothing). And, yes, someone may already have described that clearly, but they may have named it something else than "fractal Yupana". A quick search on the internet, mid november 2023, didn't show obvious results (i even tried to ask bing's ai and google's ai (bard: seem to give satisfying and precise answers in general - but without references): they didn't find anything). So... i'm trying a description of my own.
The value of a token that will be positioned in a cell on the board will be : - in/near a spot = quantity of dots in the spot multiplied by value of the cell - well outside of any spot = value of the cell (= value of its first/lowest spot)
And as in the standard interpretation of the Yupana:
- value of a column: quantity of spots in any one of its cell (or value of its first (bottom one) cell);
- value of a cell in the first row (bottom one): quantity of spots in it;
- value of a cell in a row that is not the first one: quantity of spots in it multiplied by "twice the value of the leftmost cell of the underneath row" or accumulated "value of the cell just underneath" for "each of the toes in a human" or "value of its column" counted as many times as "twice the value of the leftmost cell of the underneath row";
- value of a row: one for the first row; for every subsequent rows: value of the row just underneath counted as many times as the "quantity of fingers in a marmoset";
And, sure, that could be simpler if it was described in base 10... but we are trying to be "one, two, many - compatible", by not using any name for numbers greater than two. We should be allowed to use the words "multiply by" as it is only "accumulated as many times as", a basic concept already used in the deepest prehistory... just guessing...
"One, Two, Many - compatible" multiplication with a fractal Yupana
Basic moves
First, let's consider a few simple (or less simple) moves illustrating the rules previously stated.
If there is no error here, the quantity represented in this portion of a fractal Yupana does not change (for us it would have a name: "five hundred"), only its expression changes, given as a variety of accumulations of group of quantities... (ok, for us that means "sums of products of numbers").
Neither the meaning of the expression nor the quantity that is expressed need to be known (or named) by the user who can just follow the simple rules to correctly transform at random (or at will) the expression of the quantity. Normally, the user would want do that during a calculation in order to obtain a simpler expression (with fewer tokens, requiring less attention, minimizing the risk of errors).
Basic multiplication moves
Multiplying with a fractal Yupana is just putting elementary products (without even knowing their name or their value) on the board, and simplifying the expression when desired. The only elementary products are those produced from two of the quantities "one", "two", "one and two", "one and a pair of twos" (thethat are structuring the board and with which any number can be expressed).
Placing an elementary product a x b in the right spot of a row is a visual task:
- on that row, find a cell that has a (or b) spots in it [it has a column of value a (or b)] and where one of the spots contains b (or a) dots,
- put the token in that spot.
Illustration :
Naturally, the row on which an elementary product is placed depends on the row of each of the constituents (factors) of the product: we show a simple and "mechanical" way of managing that aspect in the following video:
Here is the same multiplication, but performed with a real Fractal Yupana + a simplified technique using less chips. The handling of the chips on the board can become very natural and quick - with some training - if the chips have a convenient shape, becoming easy to seize, push and put aside:
A curiosity...
There are so many ways you can count from 1 to 100 using a single row of the yupana... see next post, to get a feel of this gigantic quantity... Here is one way to count from 0 to 100. It uses 4 of the ways to count from 0 to 25 on the first 3 columns, and the 3 ways to represent 25 using only the last column.
The possibility of "testing on the spot" is important, because first of all, it is what helps you discover that the very notion exists.
From the experimental archaeology session in this video, we won't hesitate :-) to assert that rattlebacks were discovered, by primitive human species, millions of years ago, on exceptional beaches, just like the one pictured in the video (Crovani / Argentella beach, Corsica).
Beaches of this type were probably few in number: you need to find absolutely smooth pebbles.
But over millions of years of evolution, no doubt :-) that, multiple times, our most primitive ancestors, made this discovery, while playing on one of those beaches, during their childhood :-)
Chartres Labyrinth - How many different versions can we create ?
I recently stumbled upon a famous labyrinth in Chartres (France), 12th century.
At first sight, Chartres Labyrinth is amazing. You can't immediately figure out how it works (entrance, exit, path, dead ends, ...). It appears on the paved floor of a religious building. When I was there, the labyrinth was visible under numerous chairs installed on the floor. In those circonstances, you try to follow the path of the labyrinth with your eyes, from a distance. But its concentric circles have a puzzling visual effect. You need to follow it for real, on foot.
After a while, trying to follow the path, avoiding the chairs, it appears that it is not a maze : it has only one path, no dead ends, the entrance and the exit are the same. The specialists use the term "unicursal".
Then a question arise : how many of those labyrinths can you create, following the same design and rules (more or less). If there are more than one solution, why choose this particular one.
This question can take a bit of time.. I guess mathematicians know the answer.
I finally managed (hardship with design softwares...) to produce some variants of the Chartres Labyrinth. There seem to be a lot more...
That could be the subject of a playful hoax : flooding the internet with images of alternative versions of the Chartres Labyrinth, and see if people get caught. Am I just trying that here, on a modest scale ?
Some alternative versions to the Chartres Labyrinth - 4 sectors - 11 circuits
this one has a small dead end and a short straight path that is an island ;
not perfect, because all the surface is not used
This one has 2 small dead ends : we should avoid that
this one contains a closed curved path that is an "island" (in the middle of the down right quarter) :
an example of what we should certainly avoid
this one is better, but still has a small dead end, and a straight island
this one is cool... only 2 small dead ends
but several branches to and from the central trunk
Same version, but without the dead end
Some alternative versions, Chartres-like Labyrinths - 5 sectors - 11 circuits
Not that bad, with five lobes, one dead end here
Two small dead ends, here
Same as before, but without the dead ends,
feels a bit unbalanced ?
Chartres like Labyrinth carved on a Tagua nut - finger labyrinth - hypnoglyph
The same labyrinth as before, carved on a Tagua nut,
to be used as a finger labyrinth
Chartres like Labyrinth drawn on a spherical surface - Potato
Another 5 lobes labyrinth, wrapped onto a potato :
The same version as before, after a transposition (by different method than for the potato) on the surface of a dodecahedron.
There, each circuit/level is represented by one face of the dodecahedron, among eleven of those. The last face represents the inner and the outer space (center and outside of the Chartres-like labyrinth). Each lobe, and each of the two central paths of the Chartres-like Labyrinth (the entry and arrival at the centre) is represented by a lane in which the path will be followed.
Here a video silently illustrating (or just suggesting) the process of transposition :
Another Chartres like Labyrinth carved on a Tagua nut - finger labyrinth - hypnoglyph
The next 5 lobes, 11 circuits Chartres-like labyrinth I'll make will be this one :
here, carved on a rather angular tagua nut
Computer program generated Chartres-like Labyrinths - javascript - Google Script
June 10, 2020 - I finally wrote my own prototype of a labyrinth generator computer program... it's still imperfect, limited by my incompetence and by some restrictions of the tools i could use (exploration with Google script + rendering in html and javascript). With Google script, despite the 6 minutes limit of running time, the whole exploration is possible for 11 circuits and 5 sectors (with some pruning to avoid disgraceful labyrinths)... but not all of the output via the log (don't know how to write elsewhere, yet). For the rendering, I've some improvements to do...
Some of the 42300 labyrinths found by the computer program
See here, 1 in 150 of the more than 42300 labyrinth the computer program has found. :
See how the technique used (canevas, in html) draws labyrinths that are more "decayed" when drawn farther from the origin.
click on the video below...
More info found : what other people had been doing...
Blogmymaze - Andreas Frei
April 10, 2020 - I find a blog, BLOGMYMAZE ( https://blogmymaze.wordpress.com ), that explores labyrinths, and examines a few cases of "11 circuits, 4 sectors" labyrinths :
April 10, 2020 - Someone has already done this kind of work extensively : in 2013, Mark Wallinger, British artist, has shown 270 distinct labyrinths, one in each of London tube station. There was also a catalog of that; see for instance :
May 8, 2020 - An interesting blog compares existing (or having existed) mazes similar to the Chartres Labyrinth. It uses the rectangular transcription to ease the comparison.
Caerdroia - Labyrinthos - Andreas Frei, Hellen Galo, Tristan Smith, Jacques Hébert, ...
May 24, 2020 - Publications n°33 to n°39 of "Caerdroia - the Journal of Mazes & Labyrinths" are available in PDF versions on http://www.labyrinthos.net/digitaldownload.html. Particularly, among many other things, you'll see articles about :
- some variants of Chartres-like labyrinths, explanation from Andreas Frei about his way of cataloging labyrinths, in Caerdroia 39 (links to his website, in German, that seems rather empty) ;
- Amerindian mazes, in Caerdroia 38 ;
- how to create perfect labyrinths(Frei's graph + Hébert notation + 6 canonical labyrinths),by Ellen Galo, in Caerdroia 37 ; this article is also available in the archive of the most requested past articles... so... the subject is clearly interesting to a fair number of people :-)
- Kota labyrinths in southern India, in Caerdroia 36 ;
- how to analyze, design and scale up chartres-like labyrinths, by Andreas Frei, in Caerdroia 35 ;
- a computer program for generating medieval labyrinths, by Tristan Smith, in Caerdoia 35; we read that without enforcing strict enough rules about the properties of the labyrinths, there is ~5 million versions of a 11 circuits / 4 sectors labyrinth. The article links to Tristan Smith's website, where a good number (rapid count : more than 130) of alternative versions of Chartres-like labyrinths have been generated by the program (or with its help), using strict enough rules. The set of rules of Jacques Hébert gives just 20 "canonical labyrinths" that would be (i guess) the most "Chartres-like".
Also, the definitions corresponding to the rules that the program can respect are here https://www.otsys.com/~tsmith/properties.html(but the article in Caerdrioa is clearer : shows examples)
- a mathematical notation for medieval labyrinths, by Jacques Hébert, in Caerdroia 34;
surprise... Andreas Frei is actually also one of the authors of the blog Blogmymaze (see higher)...
The most requested past articles of Caerdoia are made available in the archive
Developing the Labyrinth: Alex Champion, p.43-51: re-drawing the classical labyrinth - new variants.
#34 (2004, pdf online)
A Mathematical Notation for Medieval Labyrinths, p.37-43: Jacques Hébert explains.
#35 (2005, pdf online)
The Cascading Serpentine, p.19-26: Andreas Frei examines the Chartres labyrinth structure.
#35 (2005, pdf online)
A Daedalus for the 21st Century, p.27-33: Tristan Smith’s software labyrinth builder.
#37 (2008, pff online)
Sigmund Gossembrot’s Labyrinth: A Very Special Design, p.41-44: Andreas Frei takes a look at an unusual labyrinth from the 15th century.
#37 (2008, pdf online)
Further Thoughts on ‘Perfect’ Labyrinths & How to Create Them, p.45-49: Ellen Galo dissects the structure of mathematically ‘perfect’ labyrinths.
#38 (2008, pdf online)
Two Labyrinths Compared: What They Have in Common, p.60-63: Andreas Frei takes a look at two apparently different labyrinths in early manuscripts.
#39 (2009, pdf online)
The True Design of Sens, p.28-32: Richard Myers Shelton compares the two known designs of the Sens Cathedral labyrinth and asks which is correct?
#39 (2009, pdf online)
A Catalogue of Historical Labyrinth Patterns, p.37-47: Andreas Frei describes the findings of his labyrinth design analysis project. Labyrinthos Archive 13
#40 (April 2011)
Greys Court: an invitation to symmetry, p.21-35: Richard Myers Shelton explores the symmetry inherent in certain labyrinths.
#40 (April 2011)
Considering the Duality of Labyrinths, p.40-47: Andreas Frei examines a hidden property of labyrinth designs.
#46 (July 2017)
Searching in the Mirror, p.34-48: Richard Myers Shelton completes his series on the geometry of symmetric labyrinth designs.
#48 (April 2019)
Basic Labyrinth Math, p.37-49: Richard Myers Shelton explains the rules of labyrinth structure.
#49 (May 2020)
Medieval Marvels: Fifty-Three Eleven-Circuit Manuscript Labyrinths, p.8-27: Jill K.H. Geoffrion & Alain Pierre Louët look at an extensive group of manuscripts produced prior to 1500
2020/09/13 - another description - with stricter rules - of what should be considered a Chartres-like labyrinth. In http://www.lavigne.dk/labyrinth/e6charst.htm. Chartres-like labyrinths would be made of a combination of those patterns :
We find various claims asserting the sun can be located, when not visible because of the clouds, with an Iceland Spar crystal. Some show experimental videos and conclude that their experiment proves that. I couldn't replicate the experiments. More precisely, I could replicate the experiment, but i couldn't see how it proved that the sun was actually located.
Conclusions of my first experiments
so far, no obvious way to tell which side is the sun ...
the sun is on the left or on the right, when "maximum" light polarization is detected
brainstorming for further tries [apparently, almost all nonsense : see "More information", lower]:
let light enter from different faces of the crystal;
really go do the experiment where the Vikings were said to use those crystals for navigation : North Sea, high latitude, on a boat ;
use refraction ; shape differently the hole that lets the light in ; use 2 crystals ;
use reflection on the inside of the faces of the crystal;
use crystals with various kind of inhomogeneities, giving complementary information ;
for example, explore the changes in colors and patterns in some crystals ;
detect the polarization from under the surface of the water;
ignore light polarization and the "Raleigh sky model", find some other model... (because the Raleigh sky model seems to have a symmetry that wouldn't allow to decide which side is the sun);
"the Viking sunstones described in the old sagas could have been dichroic cordierite, andalusite and tourmaline or birefringent calcite (Iceland spar) crystals that could serve as linear polarization analysers."
[meaning not immediately clear for the lay person !]
"— Calibration step: In cloudless weather [...] rotate (adjust) the crystal until its well-determined orientation (e.g. minimal or maximal intensity of skylight transmitted through a dichroic sunstone, or minimal or maximal intensity difference between the two slots/spots of a birefringent sunstone), where it was fixed, and thereafter he calibrated the crystal by engraving the direction pointing towards the sun on the crystal surface.
— Navigation step 1: Applying this sunstone rotational adjustment under a cloudy or foggy sky at two different celestial points, the navigator could determine the directions perpendicular to the local E-vectors of skylight shown by the engraved straight markings of the sunstones, pointing towards the sun. [?? how many sunstones are needed ??]
— Navigation step 2: The intersection of the two celestial great circles crossing the sunstones parallel to their engravings gives the position of the invisible sun." [?? celestial circles crossing the sunstones ??]
Process for calibrating an Iceland spar (calcite)
"the calcite is rotated until the intensity difference between the two light spots is maximal. This occurs four times with 90° periodicity during a full 360° rotation of the crystal. [...] the Viking navigator [...] has to scratch only one straight mark (pointing towards the sun) onto the sunstone, and this sun mark can be used under all weather conditions to determine the position of the invisible sun."
[another description, where we see that 2 sunstones are involved. And we still have to find a proposal about how you perform that on a narrow crowded ship... maybe here, if you want to pay for access : https://www.osapublishing.org/ao/abstract.cfm?uri=ao-52-25-6185 ]
"Step 1 (Fig. 1A): Viking navigators are assumed to have determined the direction of skylight polarization in at least two celestial points with the use of two sunstones to estimate the position of the sun occluded by cloud/fog or being below the horizon. [...]
Fig. 1A
Step 2 (Fig. 1B): A short scratch on each sunstone could help the navigator to set two celestial great circles across the two investigated sky points parallel to the scratches being perpendicular to the local direction of skylight polarization. Then the navigator determined the above-horizon intersection of these celestial circles. According to the Rayleigh theory of sky polarization [9], this intersection coincides with the position of the invisible sun.
Fig. 1B
Step 3 (Fig. 1C): [... hypothesis for finding the geographical north ...] "
"if you look through the crystal in its depolarizing position and then pull it away suddenly from your line of sight, you can catch a glimpse of a faint, elongate yellowish pattern known as a Haidinger's Brush. The key here is that the ends of that yellow shape point directly toward the sun."
[ The thing is not self-explanatory... there are two opposite "ends" in this elongate pattern... and they can't both point directly toward the sun... ]
The study from where this assertion comes from is here: https://royalsocietypublishing.org/doi/10.1098/rspa.2011.0369 [i'll need to try harder to understand the paper, because at first look, i can't see why we should find the direction of the sun with the Haidinger's brush... which is not directional... we find in the study this beautiful illustration, that is not as convincing as it seems to be
if you manage to see that magnificent Haidinger's brush, you can still rotate 180° and do the experiment again, then... would you not assert that the sun is in the opposite direction ? ]
Navigation by lead and line, and other means
Was it even necessary to locate the sun, the north or any direction ?
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