Thursday, November 29, 2018

algorithms for drawing celtic knots

The images above are the same celtic knot pattern (other examples here) shown in three slightly different styles. You can try creating your own knots and seeing how they look in the different styles using this online editor. All the examples above are based on the same grid:

The grid has primary points (shown in grey) and secondary points (shown in black). Breaks in the pattern are created by connecting secondary points following a few rules. The ribbon pattern is then drawn on the grid. The three styles shown are the result of three different approaches to drawing the ribbon once the grid is established.

  • The top image is created by filling in the spaces between the "ribbon" of the pattern.
  • The middle image is created by following rules that create sections of ribbon around the secondary grid points.
  • The bottom image is created by following some rules to create sections of the ribbon around the primary grid points.

All three use the idea of primary and secondary grid points, and creating the structure that defines the  patterns by joining up the secondary grid points. So the first steps of all three approaches involves setting up this basic structure. Here are those steps, adapted from this earlier post.

Setting up the grid and pattern structure

1) define primary grid points
A knot pattern is laid out on a square coordinate system using a set of "primary" points that are set at one unit distances in the horizontal and vertical directions. We'll say that (0,0) is the top left corner of the grid, and the positive x direction is towards the right and positive y direction is down.  The dimensions of the primary grid must be odd (there must be a total odd number of dots in both the x and y directions). Because we are starting with (0,0) in the top left, the top right point (x, 0) must have x even (4 in the example below), and the bottom left point (0,y) must have y even (6 in the example below).

the primary grid

2) identify secondary grid points
Some of the points on the grid are special - these form a secondary grid. The special secondary grid points are those where both x and y values are even, or both are odd.

the secondary grid

The secondary grid points where both x and y are even will be referred to as even nodes, and those that have both x and y odd will be referred to as odd nodes. The requirement to have the primary grid have odd dimensions (step 1) was needed to ensure that the corners of the pattern are all secondary points.

3) place boundaries
To create an edge-boundary for the pattern, and to create more interesting twists and turns, we follow some rules for drawing boundaries on the grid.

boundary rule 1: A boundary can connect any two non-diagonally adjacent secondary points, as long as rule 2 is not violated. The midpoint of a boundary segment will be a primary point.

boundary rule 2: A primary point cannot have more than one boundary going through it.

The example below shows boundaries drawn along the edge of the image, as well as some internal boundaries.
legal boundary examples, showing
primary and secondary points 

Three ways to draw the "knot"
Now that the grid is drawn with primary and secondary points, there are different approaches to actually creating the celtic knot pattern from the grid.

method 1: Negative space around the secondary points
To create an intertwined ribbon we can fill in the spaces around the ribbon - in addition to the boundaries already drawn, the secondary grid points are built up into polygons, and the "weaving effect" is created by drawing filaments that extend out of the secondary grid polygons, in one direction for even secondary points, the opposite for odd secondary points. 

If an adjacent primary point has a boundary through it, the line extending from the secondary point in the direction of that primary point is simply not drawn, and the corner of the polygon without a line extending from it is truncated.

A square is drawn around each secondary point, and
where there is no barrier, lines are extended based on the
even-odd rules.

Squares with missing lines can be
truncated into polygons to improve
the ribbon effect.

(Note: This post offers a slightly different description of method 1.)

method 2: Ribbon sections around the secondary points
Let's forget about the method above, and go back to the grid. Instead of filling in the spaces between the ribbon, we can draw the ribbon itself by drawing sections of the ribbon around the secondary points. Again, there is an even/odd rule in order to allow for the ribbons to pass over each other. For the node in the center, the following patterns are followed:

Adding partial paths around a secondary
node following even-odd rules

When these are joined together, the ribbon fragments form a set of continuous paths, and weave over and under each other:

portions of ribbon either join
together or appear to pass over and under

If one of the primary points surrounding the secondary point has a boundary through it, the fragments are bent and joined:

An even secondary point with a
boundary through its bottom primary neighbour.

Connecting all the lines and erasing the points and boundaries, we get something like the image below.

method 3: Ribbon sections around primary points
One final time, let's go back to the blank grid and re-draw the ribbon in a different way.

For this method, we look only at primary points that are not part of the secondary grid. This includes primary points that have (x,y) values where x is even and y is odd, or where x is odd and y is even (remember, secondary points have both x and y values even or both x and y values odd).

Like method 2, ribbon fragments are drawn, but this time the focus is on the primary nodes. again, what is drawn is different based on an "even vs odd" rule:

Lines across the primary points are drawn
using even vs odd rules

Dealing with boundaries is easier with this method, there are only two cases to consider - a vertical boundary or a horizontal boundary.

There are two cases for handling boundaries
with this method.

Any ribbon fragment that would lie outside the boundary of the grid is not drawn. Connecting all the lines and erasing the points and boundaries, we get something like the image below.

These three methods are pretty equivalent - small style alterations in any one of them can make the resulting knot look identical to one drawn using another method. When drawing knots by hand, I find that something along the lines of method 1 is easiest to use; however, from implementing each of the above in simple programs, I found that method 3 was the simplest to code.

Other Knotty Things

Other posts: 

Saturday, October 27, 2018

The unreliable guards

Here is a new puzzle variety inspired by the Forgetful Forest, Tigers and Treasure, and others by Raymond Smullyan.

Here is the setup of the puzzle:

Two guards are standing outside the entrance to a cave, guarding the treasure within. The treasure is one of copper, silver, gold, platinum, diamonds, or rubies.  
Guard 1 lies when guarding copper, silver, or gold and tells the truth when guarding other treasure. Guard 2, on the other hand, lies when guarding platinum, diamonds, or rubies, but tells the truth when guarding other treasure. 
In this land, copper is worth less than silver, which is worth less than gold, which is worth less than platinum, which is worth less than diamonds, which is worth less than rubies.

This is very similar to the Forgetful Forest, where the Lion and Unicorn each lie on particular days of the week, or in Tigers and Treasure where the inscriptions on the doors will be true only when leading to particular contents.

Just as with those puzzles, you are given clues, something like:
You meet the guards at the entrance to the treasure cave, and they make these statements: 
Guard 1 says: The treasure is more valuable than copper.
Guard 2 says: The treasure is either diamonds or rubies. 
If you determine the contents of the cave, the guards will let you pass and you can claim the treasure.
If you think you can solve this particular puzzle, try it now right here. The interactive page for this puzzle type will set you up with 838 puzzles of this variety. It presents you with a list of the treasure types, and you can select the one you think is the correct answer.

In this case, if Guard 1 says "the treasure is more valuable than copper" we can narrow down the list of possible treasures by considering two cases. In the first case, Guard 1 is telling the truth - so we know the treasure cannot be copper, silver or gold (their lying treasures); this leaves platinum, diamonds, or rubies, all of which are more valuable than copper, so the treasure could be any one of them. In the second case, Guard 1 is lying, so the treasure could be copper, silver, or gold; however, if the treasure was silver or gold, then Guard 1's statement would be true, contradicting the fact that Guard 1 is lying. So, if Guard 1 is lying, the treasure is copper, and if Guard 1 is telling the truth, the treasure is platinum, diamonds, or rubies.

Guard 1's statements provide us with a list of possible treasures: copper, platinum, diamonds, or rubies. Guard 2's statement, "the treasure is either diamonds or rubies," should narrow things down. If Guard 2 is lying then the treasure must be platinum, diamonds, or rubies (their lying treasures). However, if they are lying their statement "the treasure is either diamonds or rubies" cannot be true, so the treasure must be platinum. Looking at Guard 2's statement, there is no way it can be true because both diamonds and rubies are on Guard 2's "lying list." So Guard 2 is lying, and the only option for the treasure is platinum.

We could have solved the puzzle looking at Guard 2's statement alone: the treasure must be platinum. Because platinum is also in Guard 1's list, we can be confident that the puzzle is well-formed and that the clues are not contradicting each other.

The set of puzzles that were generated has a 'solution space' that looks like this:

The distribution shape is due to the ordering of the value of treasure types, and that we included clues that had the phrases "more valuable than" and "less valuable than" - this gave us more treasure that sat in the middle of the value range, while the ones at the ends happened less frequently. Gold and platinum satisfy the phrases "more/less valuable than x" with a greater frequency than rubies or copper.

Give these puzzles a try here. There is a Jupyter notebook here that shows how the puzzles were generated, full source for the puzzle page is here.

Illustration from Sarah Amelia Scull,
"Greek Mythology Systematized" (1880).

Thursday, September 27, 2018

some knots and not knots

Here are some knot patterns made using the online tool mentioned a few posts back. For each knot, the 'grid pattern' used to create the knot in the editor is also shown. First up, the simple unknot.

not a knot

Solomon's Knot
Next to this, we have a nice motif that is not a knot at all: the Solomon Knot is the name given to this motif of interlocked chains.

solomon unknot

BTW, you can make pattern that looks like  a solomon knot using truchet tiles:

truchet solomon unknot

Trefoil Knot
The trefoil is the simplest actual knot we can draw, but there are several ways to draw visually different but essentially equivalent trefoils, the first is considered the 'foundational' celtic knot:

trefoil 1

Here's another rendition:

trefoil 2

And one that is less recognisable:

trefoil 3

The Josephine Knot
Another non-knot (but very decorative), the Josephine consists of two interlaced links, and seems to be a favourite among crafters (who use it in belts, bracelets and macrame):

josephine link

Figure Eight Knot
The Trefoil is the only mathematically distinct prime knot with 3 crossings in a minimal planar diagram, and the figure 8 is the only one with 4. This version illustrates its name nicely:

figure eight 1

And here it is, a little lopsided:

figure eight 2

The Three-Twist
One of only two prime knots with five crossings, the three-twist (or 5_2) looks a lot like the figure 8. You can see how removing one of the lines in the grid pattern of the figure 8 allows for the additional twist that forms this knot.

the three-twist

The eight-eighteen
I don't know of a common name for the 8_18, but it's a nice looking knot, so it should have one. Here are two presentations of it:

eight-eighteen 1

eight-eighteen 2

Experiment with these and others here.

Wednesday, September 26, 2018

polynomial division practice page

I've added a new polynomial division page that generates random questions and asks you to fill in the answer, one step at a time. The page is here, along with its partners - a polynomial division calculator and example generator.

The page presents you with a randomly generated question, like this:

An initial grid is set up, with the divisor written down in the first column on the left. Everything else is unknown.

This is a grid for polynomial multiplication, but we only have one polynomial - we don't know what to put along the top for the other multiplicand. If we think of the division question as N/D = Q (numerator divided by denominator gives the quotient), the corresponding multiplication problem is DxQ = N, we have D (the denominator or divisor) and we want to find the Q to put along the top so that the contents of the grid give us N (the numerator or dividend). 

We start with the term with the highest degree in N. The page provides this prompt:

In terms of the grid, you are being asked for the coefficient for the term that will go in the leftmost empty cell in the top row.

The correct entry is -2. This allows another column of the grid to be filled in:

Now we look at the next term in N, following this prompt:

It turns out that no degree 1 term is needed in the solution since we already have all we need in the grid. When doing these by hand, you can just move on to the constant term, but on this page you actually need to put a 0 in the degree 1 column.

Moving on to the next term in N, we get this prompt:

Since there are no degree 2 terms in the table, the coefficient required is 4.

... and it turns out that the linear term also works out, as there is no remainder in this case. 

Try out the polynomial division practice page here, and visit this page for links to other posts and resources on the topic.

Tuesday, September 25, 2018

what day is it, usually?

In the Forest of Forgetfulness, Alice is trying to find out what day it is, and the unreliable denizens of the forest are not helping much. The Lion lies on Monday, Tuesday, and Wednesday and the Unicorn Lies on Thursday, Friday, and Saturday. What's more, each of them only makes one statement, and from that Alice must make her deduction.

Illustration by John Tenniel (public domain)

You can try to solve some of these puzzles here, and read about how you might solve them here.

Inspired by the original puzzles in What is the Name of this Book? by Raymond Smullyan, the Lion and Unicorn will say things like: "I told truths yesterday," or "tomorrow is one of my lying days."

If we generate a bunch of puzzles where X says "Y tells lies|truths yesterday|today|tomorrow" we end up with each creature being able to make 12 statements (after fixing the grammar a bit for verb tense). Two of those statements "I will tell truths today" (which can be said on any day), and "I will tell lies today" (which can never be said), can be thrown out,  so we get 10 statements from each creature for a combined total of 100 possible puzzles. However, it turns out that only 43 of those combinations end up generating good puzzles (puzzles where the statements lead to a unique solution). What does the set of solutions look like? We know just from the number of solutions that the 7 days of the week are not equally represented. Here's what the frequencies look like:

The first of Lion's lying days (Monday) and the last of the Unicorn's (Saturday) only occur once in the solution set, the last of the Lion's lying days (Wednesday) and the first of the Unicorns (Thursday) show up 10 times each.  The most common day is the day where they are both honest, Sunday, with 21. In this forest, it is never Tuesday or Friday.

Why not Tuesday or Friday?
This suggests a meta-puzzle: Why can Tuesday or Friday never be one of the puzzle solutions? If we look at what these days permit the Lion and Unicorn to say, we can see why this happens. Let's take Tuesday - this is a lying day for the Lion, and it happens right in the middle of Lion's lying day sequence (on Tuesday, yesterday, today, and tomorrow are all lying days). So this means the Lion can make the following statements (all lies):

I told truths today
I will tell truths tomorrow
I told truths yesterday
Unicorn told lies today
Unicorn told lies yesterday
Unicorn will tell lies tomorrow

It is a truth-telling day for the Unicorn, who can say the following:

I told truths today
I will tell truths tomorrow
I told truths yesterday
Lion told lies today
Lion told lies yesterday
Lion will tell lies tomorrow

On Tuesday, the Lion and the Unicorn can all make exactly the same statements... and also make exactly the same statements on Friday. Consequently, there is no way to distinguish between Tuesday and Friday given any pair of these statements, or tell who is lying and who is telling the truth. So none of the 36 puzzles formed by these sets of statements (the ones that could possibly be about Tuesday or Friday) are well formed puzzles (that have a unique solution).

Why is it that Sunday is so well represented in the set of puzzle solutions? Very close to half of our puzzles have Sunday as the solution (21/43). Here's an example of one of those puzzles, and here is another.

We can see how Sunday gets 21 days by listing the possible statements that can be made on that day, and seeing how the statements interact with each other. In round brackets after each statement we list all the days on which it is possible for the statement to be made (Sunday, plus some other days in some cases)

Lion's Sunday statements (all true)
I will tell lies tomorrow (Sunday, Wednesday) [5]
I told truths yesterday (Sunday, Tuesday, Wednesday, Friday) [3]

Unicorn told truths today (Sunday) [5]
Unicorn told lies yesterday (Sunday, Monday, Tuesday, Wednesday, Friday, Saturday) [3]
Unicorn will tell truths tomorrow (Sunday, Wednesday) [5]

Unicorn's Sunday statements (also all true)
I will tell truths tomorrow (Sunday, Monday, Thursday, Friday)
I told lies yesterday (Sunday, Thursday)

Lion told truths today (Sunday)
Lion told the truth yesterday (Sunday, Thursday)
Lion will tell lies tomorrow (Sunday, Monday, Tuesday, Thursday, Saturday)

Next to the Lion's statements, in square brackets we list the number of Unicorn statements that share no common days except Sunday. To Lion's statement "I will tell lies tomorrow," which can only be said on Sunday and Wednesday, all five of the Unicorn's statements can be paired to form a puzzle who's only solution is Sunday (none of the Unicorn's Sunday statements can also be made on a Wednesday). Counting up all the pairings from the list above we get 5 + 3 + 5 + 3 + 5 = 21.

Filling in some gaps

We'd like to have more puzzles, have every day represented, and not have so many Sundays. For a start, we can get some more puzzles by allowing each creature to say simply "today is Monday" or one of the other days of the week. Adding these statements increases our valid puzzle count by 67 to 110 puzzles (out of a possible 17^2 = 289 statement combinations).

Sunday more than doubles its frequency, now at 47 occurrences, and Tuesday and Friday make it in with 6 occurrences each. Monday and Saturday get 10 more occurrences, and Wednesday and Thursday get 26.

What else could we have the Lion and Unicorn say? We want them to be able to make a statement that provides a set of days as candidates for today. One example is to allow them to say "today is a weekday" or "today is the weekend." You can try out one of these puzzles here (The Lion says it is the weekend, the Unicorn says it is Friday).

Adding these two statements extend the number of statement combinations to 19^2=361, and extends the number of valid puzzles to 132. Because the 'weekend' and 'weekday' sets do not line up with the Lion and Unicorn lying days (Unicorn lies on one of the weekend days), the solution distribution is no longer symmetrical.

Monday shows up 12 times, Tuesday 6, Wednesday 38, Thursday 42, Friday 10, Saturday 13, and Sunday 54 times (down to about 41% of the puzzles).

Can we think of more statements for the Lion and the Unicorn? Sure. But with 132 puzzles to solve, let's stop here for now. Other interesting ways to change the number of solutions and their distribution is to change which days the are "truthful" for our forest friends - what if they are honest more often, and what if their lying days overlap? You can play with those questions by modifying the notebook here.