Symbolism of colors in Pagan wedding ceremonies In Pagan wedding ceremonies, colors play a significant role in symbolizing different aspects of love, spirituality, and commitment. Each color holds its own meaning and represents various emotions and energies. The use of colors in Pagan weddings allows couples to visually express their love and intentions to both themselves and their community. White, often associated with purity and new beginnings, is a common color used in Pagan wedding ceremonies. It represents the purity of love and the new chapter the couple is embarking on together. White flowers, attire, and decorations symbolize the couple's desire for a clean slate and a fresh start in their union.
The integers do not have to start with one nor do they have to be truly consecutive. A magic square can start with any number you wish and the difference between the successive set of numbers can be any common difference, that is, in arithmetic series. For instance, it could start with 15 and progress with a common difference between successive numbers of say 3. The new constant C for this type of magic square can be derived from
A magic square can start with any number you wish and the difference between the successive set of numbers can be any common difference, that is, in arithmetic series. You then proceed to place their number in the top center cell and continue to fill in the square, in the same basic pattern we just described above, telling them that when you are done, the rows, columns, and diagonals will all add up to a specific number.
White flowers, attire, and decorations symbolize the couple's desire for a clean slate and a fresh start in their union. Red, a powerful and passionate color, is frequently incorporated into Pagan weddings. It represents love, passion, and the deep connection between the couple.
Magic Squares
What do you mean "how do you make one?" If you want to know a way to construct one, my guess would be trial and error, or maybe making a simple brute-force algorithm would do it.
In an n x n square there are \(\displaystyle (n^2)!\) possibilities for a brute force approach. I'm not sure if there's a better way, although there are obvious ways to make it a bit more efficient. For example, in a 3x3, if one column already adds up to 13, you know not to put a 3 in the last square.
edit: Courtesy of wolfram, the sum of any row/column of an nxn square should be \(\displaystyle \L \frac \sum _^i = \frac n(n^2+1)\)
soroban
Elite Member
Joined Jan 28, 2005 Messages 5,586 How does one make a magic square?There are many methods available.
Each is very long to explain.
There are routines for constructing magic squares of odd order: 3-by-3, 5-by-5, etc.
There are simple rules for constructing magic squares of even order
. . if the order is a multiple of 4: 4-by-4, 8-by-8, 12-by-12, etc.
The other even orders (6, 10, 14, . ) are very tricky to construct.
Ben Franklin is said to have created a routine for these cases.
To this day, they are known as "Franklin squares".
. . \(\displaystyle \L\begin1 & 15 & 14 & 4 \\ 12 & 6 & 7 & 9 \\ 8 & 10 & 11 & 5 \\ 13 & 3 & 2 & 16\end\)
Last edited: Sep 8, 2016TchrWill
Full Member
Joined Jul 7, 2005 Messages 856 Goistein said: How does one make a magic square?Iknow Ben Franklen made one but I can't remember it.
Magic squares, those seemingly innocent looking collections of numbers that have fascinated so many for centuries, were known to the ancients, and were thought to possess mystical qualities and magical powers because of their unusual nature. In reality, they are not as magic as they are fascinating since they are usually created by following a specific set of rules or guidelines. Their creation has been a constant source of amusement for many over the years as well as studying them for their seemingly mystical properties. History records their presence in China prior to the Christian era and their introduction into Europe is believed to have occurred in the 15th century. The study of the mathematical theory behind them was initiated in France in the 17th century and subsequently explored in many other countries. The most thorough treatment of the subject may be found in the easily understood book Magic Squares and Cubes by W.S. Andrews, now published by
Dover Publications, Inc., originally published by Open Court Publishing Company in 1907. It represents the outgrowth of the famous sympoium on magic squares conducted in the Monist magazine from 1905 to 1916. It is still considered the best connected, thorough, and non-technical description and analysis of the various kinds of magic squares.
Most people are quite familiar with the most basic, and traditional, magic square where the sum of every row, every column, and the two main diagonals, all add up to a constant C. A magic square is usually referred to as a 3 cell, 4 cell, 5 cell, etc., or as a 3x3 array, 4x4 array, 5x5 array, etc. The most basic magic square of order n,
that is, n rows and n columns, or an n x n array, uses the consecutive integers from 1 to n^2. The constant sum, C, is defined by
The integers do not have to start with one nor do they have to be truly consecutive. A magic square can start with any number you wish and the difference between the successive set of numbers can be any common difference, that is, in arithmetic series. For instance, it could start with 15 and progress with a common difference between successive numbers of say 3. The new constant C for this type of magic square can be derived from
C = n[2A + D(n^2 - 1)]/2
where A is the starting integer, D is the common difference between successive terms, and n is as defined earlier.
The difference between the integers may be varied also but only between, and within, each set of n digits. By that, I mean, the series of digits can have a common difference within each set of 3 digits, and another difference between each set of 3. The only requirement that must be met is that the sum of all n^2 digits must be divisible by n^2.
Lets look at a sample series starting with 3. You could have a series such as 3-6-9-15-18-21-27-30-33. The differences between each pair in each set of three digits is 3 with the difference between the two sets of three digits being 6. Thus,
Digits. 3----6----9----15----18----21----27----30----33
Differences. 3. 3. 6. 3. 3. 6. 3. 3
The sum of the digits is 162, the row sums being 162/3 = 54. This square would look like the following:
30. 3. 21
15. 33. 6
The middle number is also the sum of the digits divided by n^2, in this case 162/9 = 18.
Some interesting characteristics of odd cell squares are:
1--The middle number of the series of numbers always goes in the middle cell.
2--The middle number is always the sum of the digits divided by n^2, or the sum of the first and last digits divided by 2, or the row sum divided by 3.
3--Any two numbers diametrically equidistant from the center add up to 2 times the center number.
ODD CELL SQUARES
Before getting into the mechanical or systematic methods of creating magic squares, let me first show you that it is possible to create one by means of simple logic, although probably only for the 3 cell square.
The usual 3 cell magic square problem is posed as, "Place the numbers 1 through 9 in the 9 cells of the 3x3 square such that each row, each column, and each diagonal add up to the same total."
Of course, the typical trial and error approach will ultimately get you to an answer but the more rewarding method is your own intuition and logic. Lets see where this takes us.
The first thing you might ask yourself is what is the total that we are seeking with the 9 digits. Since all three rows or columns must add up to the same total, it stands to reason that the sum of the rows or columns must, by definition, add up to the sum of the 9 digits, which turns out to be 45. Therefore, each row or column must add up to 45/3 = 15.
You might notice that 8 of the digits we are using just happen to add up to 10, 1+9, 2+8, 3+7, and 4+6. It might also occur to you that the middle number of the 9 digits, the 5, would most logically want to be in the middle cell with the others located around it all adding up to 15. Where to start?
What if the 9 were located on a corner? Since all three lines of numbers, including that corner 9, must toal 15, we would need three pairs of numbers that each add up to 6. Of course, this is impossible as we only have 1+5 and 2+4 at our disposal thus forcing the 9 to be located in the middle cell of one of the sides. Lets try the middle
cell of the bottom row (it could be any of the four available positions) which forces the 1 to be in the middle cell of the top row.
. 1 ?
. 5 ?
. 9 ?
Looking at our bottom row now, we notice that the two outer numbers must add up to 6 and we only have 2+4 available to us. For a reason that will become obvious later, lets try the 2 in the lower right hand corner and the 4 in the lower left hand corner.
. 1 ?
. 5 ?
. 4 9 2
What do you know? It looks like our intuition can take a rest now as the other seem to all just fall into place. The upper left cell must be an 8, the upper right must be a 6, and of course, the middle left is forced to be a 3 and the middle right becomes the 7. Here we are, and just by thinking it through.
Note that the outer numbers can be rotated clockwise or counterclocwise to define 7 additional arrangements. Mirror imaging about both vertical and horizontal axes as well as the diagonal axes will produce more. See how many different ones you can define overall.
Now we will get back to the more traditional method.
The simplest magic square has 3 cells on a side, or 9 cells altogether. We call this a three square. The simplest three square is one where you place the numbers from 1 to 9, inclusive, in each cell in such a way that the sum of every horizontal and vertical row as well as the two diagonal rows add up to the same number. This basic 3 cell square, adding up to 15, looks like the following (looks familiar):
8 1 6
3 5 7
4 9 2
This 3 cell square has some other strange characteristics. All of the four lines that pass through the center are in arithmetic progressionhaving differences of 1, 2, 3, and 4. Notice also that the squares of the first and third columns are equal, i.e., 8^2 + 3^2 + 4^2 = 6^2 + 7^2 + 2^2 = 89. The sum of the middle column squares is 1^2 + 5^2 + 9^2 = 107 which is equal to 89 + 18. The sum of the squares in the rows total 101, and 83 and, strangely enough, 101 - 83 = 18.
Other three cell magic squares can be created where the rows all add up to other numbers, the only constraint being that the sums of the rows must be divisible by 3. For instance, magics square adding up to 42 and 48 would look like this:
17 10 15 22 8 18
12 14 16 12 16 20
13 18 11 14 24 10
While we have only looked at three cell magic squares so far, you might have noticed a couple of things that turn out to be fundamental to all odd cell magic squares. First, the center square number is the middle number of the group of numbers being used or the sum of the first number and the last number divided by 2; it is also equal to the row total divided by the number of cells in the square.
Your next logical question is bound to be,"How does one create such squares? Well, I will try to explain it in words without a picture.
First draw yourself a three cell square with a dark pencil and place the numbers 1 through 9 in them as shown above. Now, above squares 1 and 6 draw two light lined squares just for reference. Similarly, draw two light lined squares to the right of squares 6 and 7. Their use will become obvious as we go along. Always place the first number being used in your square in the top center square, number one in our illustration. Now comes the tricky
part. We now wish to locate the number 2 in its proper location. Move out of the number 1 box, upward to the right, at a 45 degree angle, into the light lined box. Clearly this imaginary box is outside the boundries of our three cell square. What you do is drop down to the lowest cell in that column and place the 2. Now for the 3, move upward to the right again into the light lined box next to the number 7. Again you are outside the three cell square
so move all the way over to the left in that row and place the 3. Now you will notice that you cannot move upward to the right as you are blocked by the number 1. Merely drop down one row and place the 4 directly below the 3. Move upward to the right and place the 5. Again, move upward to the right and place the 6. You now cannot move upward to the right as there is no imaginary square there for you to move into. Merely drop down one cell and place
the 7 directly below the 6. Now move upward and to the right again and you are outside the square again. As before, merely move all the way over to the left cell in that row and place the 8. Moving upward and to the right again, you are outside the square again so merely drop down to the bottom cell in that column and place the 9. This exact same pattern is followed no matter what the rows and columns add up to.
You can also enter the numbers starting in the center box of the right column, the center box of the bottom row, or the center box of the left column as long as you follow the same pattern of locating numbers. If you were to do this you would end up with the following squares:
4 3 8. 2 9 4. 6 7 2
9 5 1. 7 5 3. 1 5 9
2 7 6. 6 1 8. 8 3 4
Moving all the outer numbers one or more boxes clockwise, or counterclockwise, also produces the same result.
Your next question is bound to be, "How does one create a magic 3 square that adds up to something other than 15?" There are two ways to create magic square for your friends. First, ask them for a number, say no more than 2 digits. You then proceed to place their number in the top center cell and continue to fill in the square, in the same basic pattern we just described above, telling them that when you are done, the rows, columns, and diagonals will all add up to a specific number.
The second way to create a magic square is to ask them for a number larger than 15 that is divisible by three. You then proceed to fill in all the cells in the same pattern such that the rows, columns, and diagonals add up to the number they gave you. Here is how you do it.
In the first method, ask them for a number, say from 1 to 25, but it can be any number. Lets say they give you 17. In your head, multiply 17 times 3 and add 12, such that 3(17) + 12 = 63. You now place the number 17 in the top center cell, continue to place 18, 19, 20, 21, 22, 23, 24, and 25 in the the cells in the same pattern as you placed the numbers 1 thru 9 above. As you are doing this, you tell them that when you are done, every row, column, and the two diagonals will add up to 63. What magic.
24 17 22
19 21 23
20 25 18
For the second method, ask them for a number that is divisible by three as you are working with a 3 x 3 square. Lets say they give you 48. In you head now, subtract 12 from the number they give you and divide the result by 3. For our example you will get 12. So place the number 12 in the top center cell and continue to fill in the other
cells in the same pattern until you reach the last cell with the number 20. Lo and behold, every row, column, and the diagonals, add up to 48.
19 12 17
14 16 18
15 20 13
You now have all the information required to create any three cell magic square possible. If anyone asks you why you use the same pattern for placing the numbers in the cells every time, fool them by rotating the pattern 90 degrees, then 180 degrees and finally 270 degrees if they really get suspicious. What this means is, for
instance, you can place the starting number in the center cell of the right most column, and so on, as I described above, and then work the same pattern starting from there. Similarly for the 180 and 270 degree rotations.
By the way, the method described above for filling in the cells is applicable to any odd cell magic square, i.e., 5, 7, etc., cell squares. The first number always goes in the top center cell. The formula for the 5 cell square is Sum = 5X + 60 where X is the number you receive from the person. If they are giving you the sum number, it must be divisible by 5 and then you subtract 60 and divide the result by 5 to get the starting number. For the 7
cell square the formula is Sum = 7X + 168. If giving you the sum number, it must be divisible by 7 and you then subtract 168 and divide the result by 7 to get the starting number. I'll leave it for you to get any others you might
be interested in from your library. The 5 cell square looks like the following:
17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9
Note that the middle number is mid way between the 1 and 25 and that the middle number is 1/5th of the row total of 65.
I'll leave the 7 cell square for you to experiment with.
A more general allocation of numbers to the cells is given by the following, given the nine numbers with n being the middle number of the nine numbers.
n + 3 n - 4 n + 1
n - 2 n n + 2
n - 1 n + 4 n - 3
CONCATENATION OF ODD CELL MAGIC SQUARES
Odd cell magic squares have another inherent characteristic derivable by concatenating (linking together) the numbers in the columns or rows. Using the basic 3 cell magic square using the digits 1 through 9, lets see what evolves.
Linking the two numbers in the first two columns, we derive the numbers 81, 35 and 49, the sum of which is 165.
Linking the two numbers in the last two columns, we derive the numbers 16, 57 and 92, the sum ofo which is 165.
Linking the two numbers in the first and last columns, we derive the numbers 86, 37 and 42, the sum of which is 165.
Performing the same linkage with the rows, we derive 83 + 15 + 67 = 165, 34 + 59 + 72 = 165 and 84 + 19 + 62 = 165.
Now, hold on to your hats.
Reversing the linkages produces
18 + 53 + 94 = 164, 61 + 75 + 29 = 165, 68 + 73 + 24 = 165, 38 + 51 + 76 = 165, 43 + 95 + 27 = 165 and 48 + 91 + 26 = 165.
Linking the numbers from any columns or rows with each other produces
88 + 33 + 44 = 11 + 55 + 99 = 66 + 77 + 22 = 88 + 11 + 66 = 33 + 55 + 77 = 44 + 99 + 22 = 165.
EVEN CELL SQUARES
Even cell magic squares are formed in an entirely different way from odd cell magic squares. Lets first examine a four cell square and its method of construction. The first thing to do is draw yourself a four cell square. Next, draw the two major diagonals of the 4 x 4 cell square, upper left corner to lower right corner and upper right corner to lower left corner. Next we are going to start moving horizontally, from box to box, starting with
the upper left most corner. We are going to count as we go along, i.e., 1-2-3-4-5. etc. As we move along from box to box, we are going to write the number we have reached in our counting in any box that does not have a
diagonal passing through it. Thus, starting with box #1, we write nothing. Moving to box #2, we write the number 2 in this second box. Likewise we write the number 3 in the third box and write nothing in box #4 as there is a diagonal passing through it. Now we drop down to the first box in the second row, directly under the box #1.
Since there is no diagonal there, we write the number 5 in this box. Boxes #6 and #7 have the diagonals passing through them so we write nothing in them. We write #8 in the eighth box. Dropping down to the first box in the third row, we write #9 in the first box, nothing in the 10th and 11th boxes, and #12 in the last box. Dropping down
to the first box in the last row, we write nothing in the first box, the #14 and #15 in the middle two boxes, and nothing in the last box. Half way there. Now, returning to the first box in the first row, we write the number 16. Counting backwards, we progress as before, left to right, row by row, writing in the empty boxes, (the ones with the diagonals passing through them) the number that we reach as we count backwards. Thus, the first box gets
#16. The next two boxes would represent #15 and #14, but there are already numbers in those boxes so we skip over them to the last box where we write the number 13. Proceeding to the next row, we skip over the #5 box, counting 12, and write the #11 and #10 in the middle two boxes of this second row , and skip over the box with the #8 in it. Down to the third row, we skip over the #9 box, counting 8, and write the #7 and #6 in the middle two boxes of
the third row, and skip over the #12 in the last box. In the last row, we write the #4 in the first box, skip the middle two boxes, and write the #1 in the very last box. We have thus created a magic square with the numbers 1 through 16 in such a way that the numbers in every row, horizontally or vertically, and the two major diagonals, all add up to 34. That wasn't too bad now, was it?
Any magic square, where the number of cells is a multiple of four, can be constructed in the same way. For instance, divide an 8 cell square into four 4 cell squares. Draw the two major diagonals in each of the four 4 cell squares just created, ending up with 8 diagonals within the four 4 cell squares. Now proceed exactly as with the 4 cell square starting with the first box in the first row. Count your way along from 1 to 64, writing the
corresponding number in any box that does not have a diagonal passing through it. Then, returning to the first box, count backwards from 64 to 1, writing the corresponding numbers in the boxes with the diagonals passing through them. You're home free. Another magic square that adds up to 260 in all the rows.
If by chance you don't feel comfortable counting backwards for the placement of the second half of the numbers, you can start at the lower right box in the last row and count forwards, moving from right to left as you move from each row to the next. It might be easier. Its up to you.
Now, as strange as it might seem, even though a 6 cell magic square is an even number square, it is not created in the manner just described. The 6 cell square is considered the most difficult magic square to create. I give you what has been advertised as the easiest method discovered to date, as far as I know.
First, divide the 6 cell square into four 3 cell squares. Starting with the upper left 3 cell square, place the numbers 1 through 9 in their proper cells exactly as you did in the regular 3 cell square. Now, moving to the lower right 3 cell square, place the numbers 10 through 18 in exactly the same pattern as you placed the 1-9 numbers.
Moving to the upper right 3 cell square, place the numbers 19 through 27 in this 3 cell square in the same pattern. And lastly, in the lower left 3 cell square, place the numbers 28 through 36 in the same pattern. Now the tricky part. Remove the numbers 8, 5, and 4 from the upper left 3 cell square, holding them aside. Move the
numbers 35, 32, and 31 from the lower left 3 cell square to the three empty spaces just created. Replace the numbers 35, 32, and 31 with the numbers 8, 5, and 4 from the cell above. Got it? You now have a 6 cell square that adds up to 111 in all the rows.
Now, as for creating a 4 cell magic square starting with a number other than one, lets explore. Ask a friend for a two digit number. Suppose he gives you 26. You immediately tell him that you will create a 4 cell magic square, starting with the number he gave you, that will add up to 134 in every row and diagonal. You determine the total by multiplying the number he gives you by 4 and adding 30, thus 4x26=104+30=134. You then proceed to fill in
the 4 cell square exactly as we described above but starting with the number 26 and ending with the number 41.
Creating a 4 cell magic square that adds up to a number given you by your friend is not as easy as it is is for an odd cell square. Obviously if your friend gives you a number greater than 34, you theoretically can work backwards from the formula for determining the total when starting with a number other than one. But since the
starting number you back into must be an integer, not any number will work. For instance, 34, 38, 42, 46, 50, 54, 58. 126, 130, 134, 138. etc. will all work, for if you subtract 30 and divide the result by 4, you get an integer number. There is no known rule for determining whether a given number will produce an integer starting
number as these numbers are not divisible by four.
FRACTIONAL MAGIC SQUARES
Fractional magic squares are possible through the same processes defined above as long as there is a constant difference between the fractions. For instance, a 3 cell magic square that adds up to 27/2 would look like this:
Another possibility is. 25/2. 1/3. 19/3
Another variation is. 29/4. 1/4. 21/4
I think you get the picture by now. Fractions less than one are possible but, again, as long as there is a constant difference between them. For example
Red decorations, garments, and accessories symbolize the intense love and desire the couple has for one another and their commitment to a passionate and fulfilling relationship. Green, the color of nature, growth, and harmony, holds its own special significance in Pagan wedding ceremonies. It represents fertility, prosperity, and the couple's desire for a strong and flourishing union. Green decorations and motifs symbolize the couple's hope for abundance and balance in their relationship, as well as their connection to the natural world around them. Blue, often associated with serenity and tranquility, is also used in Pagan weddings to symbolize harmony and peace. It represents the couple's desire for a calm and balanced relationship, free from conflict and stress. Blue decorations and clothing symbolize the couple's intention to cultivate a peaceful and harmonious union. Gold, a color associated with wealth, abundance, and divinity, is frequently incorporated into Pagan wedding ceremonies. It represents the couple's desire for a prosperous and blessed life together. Gold decorations, jewelry, and attire symbolize the couple's intention to invite abundance and divine blessings into their relationship. While these colors hold their own individual meanings, the combination of colors in Pagan wedding ceremonies can further enhance the symbolism and create a unique visual representation of the couple's love story. The use of colors allows couples to express their intentions, desires, and aspirations not only through words but also through visual symbolism, bringing an added layer of depth and significance to their wedding ceremony..
Reviews for "Unlocking the Symbolism of Dark Colors in Pagan Wedding Traditions"
1. Mark - 2/5 stars - The book "Symbolism of colors in pagan wedding ceremonies" was a great disappointment for me. I was expecting a comprehensive and insightful exploration of the use of colors in pagan weddings but instead, it felt shallow and lacking in depth. The author's analysis of colors felt repetitive and lacked any real substance. Overall, I found the book to be a missed opportunity to delve into such an interesting topic.
2. Sarah - 2/5 stars - As someone who is deeply interested in pagan ceremonies and symbolism, I was excited to pick up "Symbolism of colors in pagan wedding ceremonies." Unfortunately, I was greatly disappointed by the lack of originality and depth in this book. The author seemed to regurgitate common knowledge about color symbolism without offering any new or unique insights. It felt like a surface-level examination of the topic, and I was left wanting more profound analysis and exploration.
3. Matthew - 3/5 stars - While "Symbolism of colors in pagan wedding ceremonies" offered some interesting information about the use of colors in pagan rituals, I found the writing style to be quite dry and academic. The book lacked an engaging narrative or personal touches that would have made it more enjoyable to read. Additionally, the author's explanations often felt convoluted and hard to follow, making it a challenging book to fully grasp and appreciate. Overall, I was somewhat disappointed with this book but still found some value in its content.