The magic, myth and math of magic squares | Michael Daniels | TEDxDouglas
Summary
TLDRThe 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
🐢 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.
💠 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.
🎨 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.
📚 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.
Mindmap
Keywords
💡Magic Square
💡Lo Shu
💡Magic Constant
💡Yin and Yang
💡Fang Shu
💡Pancha Ganita
💡
💡Albrecht Dürer
💡Cornelius Agrippa
💡Sigil
💡Siamese Method
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
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