The magic, myth and math of magic squares | Michael Daniels | TEDxDouglas

Summary

The script explores the fascinating world of magic squares, dating back 4,000 years to Chinese legend, where they were believed to possess mystical powers. It delves into the Lo Shu square's significance in Chinese culture and its unique properties, such as the magic constant of 15. The script also discusses the intricate Janana square from Northern India, known as the most perfect magic square, and its diabolical intricacy with 52 ways to obtain the magic constant of 34. It touches on the historical use of magic squares for protection and in art, exemplified by Albrecht Dürer's Melencolia I engraving, which contains a magic square. The narrative concludes with the Siamese method for constructing magic squares, emphasizing their historical and mystical importance.

Takeaways

• 🔢 Magic squares are mathematical constructs where the sums of the numbers in each row, column, and diagonal are equal, known as the magic constant.
• 🐢 The concept of magic squares dates back to ancient Chinese legend, attributed to Yu the Great, who saw a pattern on the back of a tortoise, which is called the Lo Shu.
• 🧙‍♂️ Magic squares have been used in various cultures for mystical and protective purposes, such as in temples and as amulets for good luck and warding off evil.
• 🌐 The Lo Shu square is significant in Chinese philosophy, with its numbers arranged to represent the yin and yang principles.
• 💠 The 'Jana' or 'Chaisa Vatra' square is a 5x5 magic square with a magic constant of 34, notable for its intricate properties, including sums of 2x2 groupings within the square.
• 🎨 Albrecht Dürer incorporated a magic square into his engraving 'Melancholia I', which may symbolize a conflict between imagination and reason.
• 🔮 Cornelius Agrippa, a German occultist, published magic squares and attributed them to different planets, suggesting they could be used in rituals and to invoke powers.
• 🤹‍♂️ The 'Siamese method' is a simple technique to construct odd-order magic squares, which involves starting in the middle of the top row and moving up and right, with specific rules for when to move down or to the other end of a row.
• 📚 Agrippa's influence extended to figures like Dr. John Dee, and his work 'Occult Philosophy' circulated in manuscript form before its publication.
• 🤔 The speaker suggests that Dürer may have intentionally altered Agrippa's magic square in his engraving to avoid creating a pan-magic square, which might have been considered too powerful.

Q & A

What is a magic square?

A magic square is a grid of numbers where the rows, columns, and diagonals all add up to the same total, known as the magic constant.

What is the significance of the Lo Shu square in Chinese culture?

The Lo Shu square is significant in Chinese culture as it is believed to have been inspired by a pattern seen on the back of a giant tortoise. It is associated with the steps of Yu, the founder of the Xia Dynasty, and is used in religious practices and architecture for protection.

What is the magic constant of the Lo Shu square?

The magic constant of the Lo Shu square is 15, which is the sum of the numbers in each row, column, or diagonal.

How does the arrangement of numbers in the Lo Shu square relate to Chinese philosophy?

In the Lo Shu square, the odd (yin) numbers are placed in the corners, and the even (yang) numbers form a cross in the center, reflecting the balance of passive and active forces in Chinese philosophy.

What is a pan-magic square?

A pan-magic square is a type of magic square where not only do the rows, columns, and diagonals add up to the same total, but also every 2x2 grouping within the square adds up to the magic constant.

What is the significance of the number 34 in the Jana (Chia-Sa-Vatra) square?

In the Jana square, the number 34 is the magic constant, and it is significant because it can be obtained in 52 different ways from the square, including the sums of the 2x2 groupings.

What is the connection between the magic square in Albrecht Dürer's engraving 'Melencolia I' and Cornelius Agrippa?

Albrecht Dürer's magic square in 'Melencolia I' is believed to have been inspired by Cornelius Agrippa, a German occultist. Agrippa's influence on Dürer is evident in the engraving, which includes a magic square similar to one published by Agrippa.

Why is it impossible to create a 2x2 magic square?

A 2x2 magic square is impossible because each number in the square must be unique, and with only four cells, it's not possible to have different sums for the rows, columns, and diagonals.

What is the Siamese method for constructing odd-order magic squares?

The Siamese method for constructing odd-order magic squares involves starting in the middle of the top row, moving up and right, and if you cannot move up and right, you move to the end of the row and then down. If you move out of the square, you go to the opposite end of the row.

How were magic squares used in the past?

Magic squares have been used for various purposes in the past, including as protective symbols in buildings, as amulets for personal protection or healing, and in rituals to invoke the powers of the planets or summon supernatural beings.

Outlines

00:00

🐢 Ancient Legends and the Origins of Magic Squares

The first paragraph introduces the narrator's fascination with magic squares and connects their history to Chinese mythology. It recounts the tale of the great Yu from the Xia dynasty, around 2000 BC, who devised a magic square after seeing a pattern on a tortoise’s back, known as the 'Lo Shu.' This grid, featuring the numbers 4, 9, 2, 3, 5, 7, 8, 1, 6, is symbolic in Chinese religion and philosophy. The paragraph also explains how Taoist priests used the pattern in their ritualistic steps to gain supernatural powers. Importantly, the square's magic lies in the fact that all rows, columns, and diagonals add up to 15, representing a 'magic constant' with further significance in the Chinese calendar.

05:01

💠 The Diabolic Perfection of the Pan Magic Square

This paragraph delves into the intricacies of the Jain Square, also known as the Chautisa Yantra or Diabolic Square, which is considered a 'pan magic square' due to its highly complex properties. Not only do the rows, columns, and diagonals sum up to 34, but every 2x2 grouping within the square also equals 34. The paragraph explores how there are 52 ways to obtain this constant, highlighting its ingenious design. The term 'pan magic' refers to how numbers spaced two units apart on the diagonals sum to 17, half of the magic constant. Additionally, tracing the numbers in sequence reveals a fascinating pattern, making this square a remarkable feat of mathematical symmetry.

10:03

🎨 Albrecht Dürer’s Enigmatic Magic Square

The third paragraph shifts focus to the famous German artist Albrecht Dürer and his 1514 engraving 'Melencolia I,' which features a 4x4 magic square. This square, unlike the pan magic square, doesn't have all diagonals summing to 17, but it does possess interesting properties, such as the rows, columns, diagonals, and the four corners totaling 34. Dürer’s square may have been influenced by the German occultist Cornelius Agrippa, and a simple transformation can make it pan magic. The connection between Dürer’s magic square and Agrippa’s work suggests that Dürer intentionally altered the square, perhaps embedding personal significance, like the number '1514' in the engraving's bottom row.

15:05

📚 Agrippa's Occult Influence on Magic Squares

This paragraph elaborates on Cornelius Agrippa, a major influence on Dürer. Agrippa’s 'Occult Philosophy,' published in the early 1500s, included various magic squares attributed to planets, with each square believed to encapsulate astrological powers. Agrippa's 3x3 square, for example, corresponds to Saturn and matches the earlier Lo Shu square. The paragraph explains how these squares were used in rituals to invoke planetary power or even summon supernatural entities. Agrippa's reluctance to publish the pan magic square hints that he may have considered its power too dangerous or 'diabolical,' a concept central to Renaissance-era occult thought.

🔮 Constructing Your Own Magic Squares

The final paragraph focuses on the construction of magic squares, particularly the difference between odd- and even-order squares. Odd-order squares are simpler to create using the 'Siamese method,' a process that originated in Thailand. Starting from the top middle and moving up and to the right, the pattern fills itself, with specific rules to handle when you move out of bounds. By following these rules, anyone can create a basic magic square, providing a fun trick to impress others while also respecting the inherent mysticism of these mathematical marvels.

👏 Conclusion and Audience Applause

The final section contains the concluding moments of the video script, with the presenter wrapping up their explanation and the audience offering applause. The presenter reminds viewers to respect the mystical nature of magic squares, reiterating their magical and mathematical significance.

Keywords

💡Magic Square

A magic square is a grid of numbers where the sums of the numbers in each row, column, and diagonal are the same. This is a central concept in the video, which explores the history and significance of magic squares. The video mentions the Lo Shu square, a 3x3 magic square originating from Chinese legend, which has a magic constant of 15. This square is used to illustrate the defining characteristic of magic squares.

💡Lo Shu

The Lo Shu is a specific 3x3 magic square mentioned in the video, which is believed to have been devised around 2000 BC by the great Yu. It is significant in Chinese philosophy and religion, with the numbers arranged to have symbolic meanings related to yin and yang. The video uses the Lo Shu square to demonstrate the magic square's properties and its cultural importance.

💡Magic Constant

The magic constant is the sum of the numbers in any row, column, or diagonal of a magic square, which is the same for all rows, columns, and diagonals. The video explains that in the Lo Shu square, this constant is 15, which is also the number of days in each cycle of the Chinese calendar, highlighting the magic constant's symbolic significance.

💡Yin and Yang

Yin and Yang are opposing principles in Chinese philosophy that represent, respectively, passive and active forces. The video describes how the Lo Shu square's arrangement of odd and even numbers reflects these principles, with even numbers (yin) in the corners and odd numbers (yang) forming a cross in the center, demonstrating the integration of magic squares with philosophical concepts.

💡Fang Shu

Fang Shu refers to Chinese geomancy or the practice of siting and arranging buildings to achieve harmony with the environment. The video mentions that the arrangement of numbers in the Lo Shu square has great significance in Fang Shu, indicating the use of magic squares in architectural design for achieving balance and harmony.

💡Pancha Ganita

Pancha Ganita, or 'fivefold mathematics,' refers to a classification of mathematical knowledge in ancient India. The video discusses a particular magic square known as the Jana or Chaisa Vatra, which is an example of a highly intricate magic square with multiple layers of symmetry and sum properties, illustrating the advanced mathematical concepts within Pancha Ganita.

💡

💡Albrecht Dürer

Albrecht Dürer was a famous German artist mentioned in the video for his engraving 'Melancholia I,' which features a magic square. Dürer's work is discussed to explore the cultural and artistic significance of magic squares, suggesting that they were used not only for mathematical curiosity but also as symbols in art and philosophy.

💡Cornelius Agrippa

Cornelius Agrippa was a German occultist whose work is connected to Dürer's magic square in 'Melancholia I.' The video suggests that Dürer may have obtained his magic square from Agrippa's manuscripts, indicating the influence of occultism and the exchange of mathematical knowledge through personal networks.

💡Sigil

A sigil is a symbol used in magic to represent ideas or invoke powers. The video mentions that magic squares were used to create sigils for summoning or invoking supernatural entities, showing the practical application of magic squares in esoteric practices.

💡Siamese Method

The Siamese Method is a technique for constructing odd-order magic squares, which the video explains through a simple step-by-step process. This method is highlighted as an accessible way to create magic squares, emphasizing the practical and recreational aspects of magic squares beyond their cultural and symbolic significance.

Highlights

Magic squares have fascinated people for thousands of years, combining elements of magic, myth, mystery, imagination, and mathematics.

The Lo Shu square, originating from Chinese legend, is one of the earliest known magic squares, with a history possibly dating back 4,000 years.

The Lo Shu square is believed to have been inspired by the pattern on the back of a giant tortoise emerging from a flooded river.

The numbers in the Lo Shu square have significance in Chinese religion and magic, with the sum of numbers in each row, column, and diagonal being 15, known as the magic constant.

In Chinese philosophy, the placement of odd (yin) and even (yang) numbers in the corners and center of the square holds symbolic meaning.

The magic constant of 15 is significant as it represents the number of days in each cycle of the Chinese calendar.

Magic squares are not possible with two rows and columns due to the requirement for unique numbers, but can be created for any higher order.

The Jana square, also known as the Chaisa Vantra, is a particularly complex and perfect magic square with a magic constant of 34.

The Jana square is unique as it has 52 different ways to obtain the magic constant, including sums of 2x2 groupings within the square.

The square is also referred to as pan magic or uber magic due to its intricate properties, including the sum of numbers spaced two apart on diagonals being half the magic constant.

Magic squares are often found in temples and buildings, believed to provide magical protection and ward off evil.

Magic squares have been used as amulets for protection, healing, and good luck throughout history.

Albrecht Dürer's engraving 'Melancholia I' contains a magic square, which has been a subject of much debate and interpretation.

Dürer's magic square is not pan magic, but it has additional features such as the four corner squares and central squares both summing to 34.

Cornelius Agrippa, a German occultist, is believed to be the source of Dürer's magic square, with his work 'Occult Philosophy' possibly influencing the engraving.

Agrippa's own magic squares were attributed to different planets, encapsulating astrological qualities and used in rituals.

The Siamese method, originating in Thailand, is a simple technique for constructing odd-order magic squares.

Magic squares should be respected for their magical properties, as they have been revered throughout history.

Transcripts

I've been interested

in magic squares since my early teens

and if I can I'd like to share some of

that fascination with you today because

it's a story of magic and myth and

mystery and Imagination and Mathematics

but I'm going to try and keep the math

to a minimum and I also stop there with

the

alliteration and it's a story that goes

back perhaps 4,000 years and it all

begins with

turtles according to Chinese Legend the

great Yu who was the founder of the jar

Dynasty around 2000

BC devised this pattern which he saw on

the back of a giant tortoise as it

emerged from the flooded of the the

river low and this is called the low Sho

which means the river

writing and if you count the dots in

each segment of the picture you'll see

there are numbers there 492 357

816 and these have very important

significance in Chinese religion and

Magic one of the uses of this Lou Square

relates to something called the steps of

U or the

yubo the DST priests have a Mystic Dan

step which is to step forward and then

slide the back foot and that's because

you himself was lame through all his

labors and if you trace the numbers of

that Square from 1 to 2 to 3 to four and

so on you create that pattern and and

the DST priests will walk through the

temple using that movement from one to

two and by entering the different areas

of the temple Each of which has its own

particular significance they believe

that they acquire Supernatural

Powers but this particular square has

other interesting symbolic features

you'll note that the odd

numbers the yin numbers the passive

numbers in Chinese philosophy are in the

corner ERS and the odd

numbers the Yang numbers the active

numbers form a cross in the center and

that has great significance particularly

in

Fang but the thing about this square

that makes it

magic is the fact that if you add up the

numbers in each row or the numbers in

each column or the numbers on both

diagonals you get the same total which

is

15 and that's what makes it a magic

Square that's the defining

characteristic of a magic square that

the rows the columns and the two

diagonals will add up to the same number

and that number is called the magic

constant and the number 15 also has

symbolic significance in Chinese

philosophy because it's the number of

days in each cycle of the Chinese

calendar now you can create magic

squares for any number of rows and

columns except two you can't have a 2 by

two magic square because each number has

to be different so you can't simply

write two in each Square because well

that wouldn't be magic but you can

create magic squares for any higher

order that you wish and using

computers squares have been created for

huge numbers of rows and columns but I

want to focus on this particular Square

for a moment it's a fascinating and

extremely important Square it's found in

this temple in Northern India it's a

Jane Temple

and magic squares often appear in

buildings and it seems like they're

there as a kind of magical protection to

ward off evil for the same reason

throughout history people would wear

magic squares around their necks as

amulets or perhaps they would have them

embroidered on their clothes for Magical

protection or for healing or simply for

good

luck but this particular square the

janaa square also known as the chaisa

vantra is particularly interesting in

fact it's been called diabolic a

diabolic magic square not because it's

anything to do with the devil but

because it is diabolically intricate and

ingenious because not only do the rows

the columns and the diagonals add up to

the same total which in this case is

34 but every 2x two grouping within that

square adds up to

34 the four corner squares add up to 34

the two diagonal squares in opposite

Corners add up to

34 in fact there are

52 different ways that we can obtain the

magic constant from this particular

Square it is truly diabolic it's also

called pan magic or Uber magic or most

perfect it is the most perfect magic

square that is

possible so what makes it pan magic well

it's all to do with the

diagonals there is a un Universal

feature on the diagonal which is that if

you look at any two numbers which are

spaced two apart on either diagonal they

add up to 17 which is exactly half the

magic constant and it's that feature

that makes this a pan magic square and

also they're very interesting design

that you can obtain if again if you

trace the numbers 1 2 3 4 5 all the way

through the square you obtain this quite

interesting complex not quite

symmetrical pattern now this Square the

Jana Square as it's called dates to

around a

thousand the year a thousand of the

Common Era our Common Era I want to jump

forward now 500 years to this man alre

Durer famous German artist who in 154

created this interesting and enigmatic

engraving it's entitled Melancholia 1 or

perhaps it's Melancholia

I and this has been the subject of much

discussion and debate for many years and

interpretations of the engraving vary

widely but perhaps the most likely

explanation is that it's a reference to

the first type of Melancholia that was

identified

by the German occultist Cornelius

agria Agrippa talks about Melancholia

imagina or Melancholia of the

imagination and he believed it

particularly affected

artists and it was due to a conflict

between

imagination and reason and you can see

features of that in the in the engraving

the figure there looks rather foror has

wing

is surrounded by various instruments of

science there's a very strange

polyhedron in the picture on the left

which actually is unique in art history

it's known as the duror

solid in the

background there are more imaginative

features perhaps the Sun the sea and a

rainbow but I want to draw your

attention to the upper right just below

the Bell because there is yes a magic

square and there it is now the first

thing to notice is that this is not a

pan magic magic square and we can tell

that because we don't have the numbers

on the diagonals adding up to 17 if you

take numbers that are two apart so this

is not a pan diagonal a pan diagonal pan

magic square but it is an interesting

Square because it has additional

features as well as the rows and the

columns and the diagonals the four

corner squares will add up to

34 and the four Central squares add up

to 34 and it does create quite an

interesting

pattern now the question that is begged

here I think is where did diur get this

magic Square from and the answer is this

man Cornelius agria

again but one thing I want to show is

that

the dura

Square can be made pan magic very very

easily if you take the dura square and

you

swap the bottom two rows and then you

swap the right two columns you obtain

this panm magic

square because now the two numbers to

apart on the diagonals do add up to 17

in every case and so that would produce

a pattern whereby you could obtain the

magic constant of 34 in these 52

different ways and also it's interesting

if you have a look at the pattern that

it now makes it's symmetrical about the

horizontal midline perfectly symmetrical

it's a very interesting square but the

one that Jura published was not pan

magic the one that AG gripper

published is also not pan magic AG

gripper was a very famous occultist he

influenced all sorts of people he

influenced Dr John D the court

astrologer and adviser to Elizabeth I

for example and his most famous work is

this it's called Occult Philosophy and

it was published in 1531 originally and

then over the next couple of years but

it had circulated privately in

manuscript form for many years before

that and it's very likely indeed I think

it's certain

that jurer obtained a copy of a

gripper's manuscript and hence he

obtained his own

Square because the two squares that a

grippa publishes and the one that Jura

has in his engraving are variations very

simple transformations of each other if

you take a grippa 4x4 square that he

published and you invert it and then you

swap the middle two columns then you

obtain jura's Square so why did Jura do

that why did he transform a gri of

square I think the reason lies in the

bottom row so if you look at durus

Square you'll note that the the two

middle numbers are 154 which is the date

of the

engraving there's another interesting

question that is begged by this square

and that is because it's so easily

transformed into a pan magic square

why didn't a gripper publish it did he

know of the pan magic variation I think

he did and I think he thought that the

the magic was to just too powerful too

diabolic and he decided to keep it

secret now Agrippa also published other

magic

squares of different sizes and he

attributes these to the different

planets the 3X3 square is attributed to

Saturn and that is exactly the same as

the lotion Sho square that we saw

earlier and the other squares are

attributed to the other planets they

were called planets they were known as

planets at the time because this was

about 10 years before

cernus and it was believed that each of

these squares encapsulated the

astrological qualities and powers of

each of those planets and they could be

used in rituals for example

to bring down the power of Venus or to

bring down the power of the Moon you

would use that particular magic

square they were also used to create

sigils for the summoning of demons or

the invocation of angels using a very

specific procedure which I don't have

time to go into but what I will go into

because you might find this interesting

is how can we create magic

squares one thing to notice is the

designs that are traced out by the seven

a group of squares and I'd like you to

notice the difference between the odd

order squares and the even order

squares the odd order squares have a

very symmetrical and regular and simple

pattern the even numbered squares are

much more irregular in their structure

and that relates to the way that we can

construct these magic squares it's very

difficult to construct an even order

Magic Square because they are extremely

intricate the odd automatic squares are

actually very easy to construct and

perhaps you'd like to learn how to do it

so that you can impress your friends it

makes a nice party trick it's a very

simple method it came to the West in the

17th century it's called the Siamese

method because it originated in sayam

Thailand as it now is you start in the

middle at the top and then you move up

and right and if you can't move up and

right you move to the end end of the row

that you would have moved into so you

start with

one you would move out of the square so

you end up going to the other end of

that particular row so

two will go

there three will go there because you go

upright you then move out of the square

so you would then move to the other end

of that Row the four would go there you

would go upright five the other rule is

if you get stuck you move down so you

then go down six up

seven up

eight and so on and we can carry all the

way through using those very simple

rules and we

create the magic

square which has a total of

25 and that's something about magic

squares I hope you find them interesting

I hope you enjoy the game but remember

also that magic squares should be

respected because they are after all

magic thank

[Applause]

you