Discretization and tropicalization: How are they related? Christer - - PowerPoint PPT Presentation
Discretization and tropicalization: How are they related? Christer O. Kiselman Member of the Reference Group for Mathematics of ISP First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm, Sida Headquarters 2017
Member of the Reference Group for Mathematics of ISP
Stockholm, Sida Headquarters 2017 March 07 15:00–15:45
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics.
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics. I will present them and motivate why they are important.
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics. I will present them and motivate why they are important. Discrete objects, like carpets and mosaics, have been around for thousands of years, but the advent of computers and digital cameras has made them ubiquitous.
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics. I will present them and motivate why they are important. Discrete objects, like carpets and mosaics, have been around for thousands of years, but the advent of computers and digital cameras has made them ubiquitous. Tropical mathematics is by comparison a relatively new branch
Abstract Discretization and tropicalization are two important procedures in contemporary mathematics. I will present them and motivate why they are important. Discrete objects, like carpets and mosaics, have been around for thousands of years, but the advent of computers and digital cameras has made them ubiquitous. Tropical mathematics is by comparison a relatively new branch
I shall compare the two and also pose a philosophical/mathematical problem in relation to them.
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Contents
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Carpets, embroidery work, and mosaics are examples of artifacts that are discrete.
Carpets, embroidery work, and mosaics are examples of artifacts that are discrete. A carpet can consist of thousands of knots, but they are only finite in number.
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Persian carpet.
A mosaic is made up of finitely many little stones, tessellas.
11
Ravenna mosaic.
12
Embroidery ...
13
All three can present pictures quite well.
All three can present pictures quite well. These objects have been around for a long time, but now, with computers and digital cameras, they are everywhere. A photo consists of many pixels (picture elements) but only finitely many.
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So all this makes digital geometry into a kind of geometry of growing interest. Discretization of sets and function is studied from many points of view.
So all this makes digital geometry into a kind of geometry of growing interest. Discretization of sets and function is studied from many points of view. Tropicalization is somehow similar to discretization ...
So all this makes digital geometry into a kind of geometry of growing interest. Discretization of sets and function is studied from many points of view. Tropicalization is somehow similar to discretization ... Here I will talk about both procedures.
18
Intuitively, a discrete set is a set where every point is a bit away from all the other points.
Intuitively, a discrete set is a set where every point is a bit away from all the other points. To make this precise, we need some kind of distance. So denote by d(x,y) the distance between two points x and y in some kind of space, like the plane or three-space.
Intuitively, a discrete set is a set where every point is a bit away from all the other points. To make this precise, we need some kind of distance. So denote by d(x,y) the distance between two points x and y in some kind of space, like the plane or three-space. Then a set A is said to be discrete if for every element a of A, there is a positive number r such that d(a,b) > r for all elements b 6= a of A.
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Let us take R as an example, with d(x,y) = |x y|. Then for every number a there are other numbers b that are as close as we like to a. So R with this metric is not discrete.
Let us take R as an example, with d(x,y) = |x y|. Then for every number a there are other numbers b that are as close as we like to a. So R with this metric is not discrete. The subset Z of integers is discrete, because now d(a,b) > 1 for all points b 6= a.
Let us take R as an example, with d(x,y) = |x y|. Then for every number a there are other numbers b that are as close as we like to a. So R with this metric is not discrete. The subset Z of integers is discrete, because now d(a,b) > 1 for all points b 6= a. But beware! Every set is discrete if we define the distance as
d(x,y) = 1 when x 6= y and d(x,x) = 0. So also R is discrete
with this kind of distance.
Let us take R as an example, with d(x,y) = |x y|. Then for every number a there are other numbers b that are as close as we like to a. So R with this metric is not discrete. The subset Z of integers is discrete, because now d(a,b) > 1 for all points b 6= a. But beware! Every set is discrete if we define the distance as
d(x,y) = 1 when x 6= y and d(x,x) = 0. So also R is discrete
with this kind of distance. This means that the distance between π and 3.141592653589793238462643383279502884197 is 1 with this metric, whereas it is rather small in the unusal metric.
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Mathematical models based on real or complex numbers have been very successful—think of celestial mechanics and the theory of electric circuits.
Mathematical models based on real or complex numbers have been very successful—think of celestial mechanics and the theory of electric circuits. However, we should not think of real numbers as being more real than complex numbers or than discrete models. The attribute real is misleading.
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That time is discrete was a theory developed in Spain in the Middle Ages:
That time is discrete was a theory developed in Spain in the Middle Ages: “As regards the theoretical and philosophical analysis of time, the most important and original contribution of medieval Islamic thinkers was their theory of discontinuous, or atomistic, time.” (Whitrow 1990:79)
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Moshe ben Maimun (born in 1135 or 1138 in C´
deceased in 1204 near Cairo), a Jewish philosopher working in Muslim cultural circles and better known under his Greek name Maimonides, Ma¨ imwn–dhc, wrote:
Moshe ben Maimun (born in 1135 or 1138 in C´
deceased in 1204 near Cairo), a Jewish philosopher working in Muslim cultural circles and better known under his Greek name Maimonides, Ma¨ imwn–dhc, wrote: “Time is composed of time-atoms, i.e. of many parts, which in account of their short durations cannot be divided. ... An hour is, e.g. divided into sixty minutes, the second into sixty parts and so on; at last after ten or more successive divisions by sixty, time-elements are obtained which are not subject to division, and in fact are indivisible.” (Whitrow 1990:79)
Moshe ben Maimun (born in 1135 or 1138 in C´
deceased in 1204 near Cairo), a Jewish philosopher working in Muslim cultural circles and better known under his Greek name Maimonides, Ma¨ imwn–dhc, wrote: “Time is composed of time-atoms, i.e. of many parts, which in account of their short durations cannot be divided. ... An hour is, e.g. divided into sixty minutes, the second into sixty parts and so on; at last after ten or more successive divisions by sixty, time-elements are obtained which are not subject to division, and in fact are indivisible.” (Whitrow 1990:79) Often we do not go from models based on real numbers to discrete models, but from one very fine discrete model to another model, also discrete, but less fine.
Moshe ben Maimun (born in 1135 or 1138 in C´
deceased in 1204 near Cairo), a Jewish philosopher working in Muslim cultural circles and better known under his Greek name Maimonides, Ma¨ imwn–dhc, wrote: “Time is composed of time-atoms, i.e. of many parts, which in account of their short durations cannot be divided. ... An hour is, e.g. divided into sixty minutes, the second into sixty parts and so on; at last after ten or more successive divisions by sixty, time-elements are obtained which are not subject to division, and in fact are indivisible.” (Whitrow 1990:79) Often we do not go from models based on real numbers to discrete models, but from one very fine discrete model to another model, also discrete, but less fine. Physicists study discrete models of spacetime ... and not just recently—but much later than Maimonides ...
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To discretize To discretize a set means to map it in some way to a discrete set.
To discretize To discretize a set means to map it in some way to a discrete set. A simple example of a discretization is the mapping P(Rn) 3 A 7! A\Zn 2 P(Zn), mapping any subset of Rn to the set of its points with integer coordinates.
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Since A may lack points with integer coordinates, we often want to fatten the set before we intersect it with Zn: P(Rn) 3 A 7! (A+ U)\Zn 2 P(Zn), where U can be any set; most typically a ball or cube to guarantee that the image is nonempty as soon as A is nonempty.
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We can discretize a function f : R ! R partially or completely:
We can discretize a function f : R ! R partially or completely: We may just take its restriction to the integers, i.e., f|Z, and keep its values in R.
We can discretize a function f : R ! R partially or completely: We may just take its restriction to the integers, i.e., f|Z, and keep its values in R. Or we may discretize also its values, e.g., by taking
g(x) = bf(x)c, x 2 Z.
We can discretize a function f : R ! R partially or completely: We may just take its restriction to the integers, i.e., f|Z, and keep its values in R. Or we may discretize also its values, e.g., by taking
g(x) = bf(x)c, x 2 Z. Here btc denotes the floor function,
defined by R 3 t 7! btc 2 Z and t 6 btc < t + 1.
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2 4 6 8 10 12 14 16 18
1 2 3 4 5 6
Fonction plancher Droite euclidienne
Covering the Euclidean straight line of equation y = 1
3x by a
dilation with structural element equal to the rectangle [1
2, 1 2]⇥[5 6, 5 6] (courtesy Adama Arouna Kon´
e).
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Covering a Euclidean plane by a dilation with structural element equal to the box [1
2, 1 2]⇥[1 2, 1 2]⇥[9 8, 9 8] (courtesy
Adama Arouna Kon´ e).
Small children start to count with natural numbers. Real numbers are much more difficult to define.
Small children start to count with natural numbers. Real numbers are much more difficult to define. So one may think that discrete sets, like the set of natural numbers N = {0,1,2,3,...}, are easier to handle than the set R
Small children start to count with natural numbers. Real numbers are much more difficult to define. So one may think that discrete sets, like the set of natural numbers N = {0,1,2,3,...}, are easier to handle than the set R
However, ...
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In A decision method for elementary algebra and geometry, Alfred Tarski (1901–1983) showed that the first-order theory of the real numbers under addition and multiplication is decidable. Via Descartes (1596–1650), this applies to elementary Euclidean geometry.
In A decision method for elementary algebra and geometry, Alfred Tarski (1901–1983) showed that the first-order theory of the real numbers under addition and multiplication is decidable. Via Descartes (1596–1650), this applies to elementary Euclidean geometry. (This result was published only in 1948, but it dates back to 1930 and was mentioned in Tarski (1931).)
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This is a most remarkable result, because Alonzo Church (1903–1995) proved in 1936 that Peano arithmetic (the theory
This is a most remarkable result, because Alonzo Church (1903–1995) proved in 1936 that Peano arithmetic (the theory
Peano arithmetic is also incomplete by G¨
theorem (Kurt G¨
This is a most remarkable result, because Alonzo Church (1903–1995) proved in 1936 that Peano arithmetic (the theory
Peano arithmetic is also incomplete by G¨
theorem (Kurt G¨
So, all this points to the fact that, from the point of view of logic, natural numbers are much more difficult to treat than real numbers.
This is a most remarkable result, because Alonzo Church (1903–1995) proved in 1936 that Peano arithmetic (the theory
Peano arithmetic is also incomplete by G¨
theorem (Kurt G¨
So, all this points to the fact that, from the point of view of logic, natural numbers are much more difficult to treat than real numbers. But not only logic!
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The derivative of f(x) = x5, x 2 R, is f 0(x) = 5x4.
The derivative of f(x) = x5, x 2 R, is f 0(x) = 5x4. But the difference quotient is
f(x + h) f(x) h
= 5x4 + 10x3h + 10x2h2 + 5xh3 + h4.
The derivative of f(x) = x5, x 2 R, is f 0(x) = 5x4. But the difference quotient is
f(x + h) f(x) h
= 5x4 + 10x3h + 10x2h2 + 5xh3 + h4. And coming to integrals we have to compare
Z a f(x)dx = 1 6a6
and
1 n
an
j=0
f(j/n) = 1 6a6 + 1 2 a5 n + 5 12 a4 n2 1 12 a2 n4.
The derivative of f(x) = x5, x 2 R, is f 0(x) = 5x4. But the difference quotient is
f(x + h) f(x) h
= 5x4 + 10x3h + 10x2h2 + 5xh3 + h4. And coming to integrals we have to compare
Z a f(x)dx = 1 6a6
and
1 n
an
j=0
f(j/n) = 1 6a6 + 1 2 a5 n + 5 12 a4 n2 1 12 a2 n4.
This may explain some of the success of differential and integral calculus.
55
Balayage is a method developed in classical potential theory for measures (masses or electric charges).
Balayage is a method developed in classical potential theory for measures (masses or electric charges). The term is of French origin and means ‘sweeping (using a broom)’.
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In this context we may define the balayage of a function f defined in Rn onto a discrete set P as the function
g(p) = ∑
a2V(p)
f(a), p 2 P,
where the sets V(p), p 2 P, form a tessellation of Rn, meaning that
V(p)\ V(q) = Ø when p 6= q, and [
p2P
V(p) = Rn.
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We may for example take P = Zn ⇢ Rn and
V(p) = {a 2 Rn; bac = p}, p 2 Zn,
We may for example take P = Zn ⇢ Rn and
V(p) = {a 2 Rn; bac = p}, p 2 Zn,
V(p) = {a 2 Rn; ba+(1
2, 1 2,..., 1 2)c = p},
p 2 Zn.
We may for example take P = Zn ⇢ Rn and
V(p) = {a 2 Rn; bac = p}, p 2 Zn,
V(p) = {a 2 Rn; ba+(1
2, 1 2,..., 1 2)c = p},
p 2 Zn.
The Voronoi cells are named for Georgi˘ ı Feodoseviˇ c Vorono˘ ı Georg⇢ Feodos⇢oviq Voroni⇢; Georgi⇢ Feodos~eviq Vorono⇢, Georges Vorono¨ ı (1868–1908).
We may for example take P = Zn ⇢ Rn and
V(p) = {a 2 Rn; bac = p}, p 2 Zn,
V(p) = {a 2 Rn; ba+(1
2, 1 2,..., 1 2)c = p},
p 2 Zn.
The Voronoi cells are named for Georgi˘ ı Feodoseviˇ c Vorono˘ ı Georg⇢ Feodos⇢oviq Voroni⇢; Georgi⇢ Feodos~eviq Vorono⇢, Georges Vorono¨ ı (1868–1908). Given a set P of points in a space X with distance d, we define the Voronoi cell with kernel p 2 P as the set
V(p) = {x 2 X; d(x,p) 6 d(x,q) for all points q 2 P}.
We may for example take P = Zn ⇢ Rn and
V(p) = {a 2 Rn; bac = p}, p 2 Zn,
V(p) = {a 2 Rn; ba+(1
2, 1 2,..., 1 2)c = p},
p 2 Zn.
The Voronoi cells are named for Georgi˘ ı Feodoseviˇ c Vorono˘ ı Georg⇢ Feodos⇢oviq Voroni⇢; Georgi⇢ Feodos~eviq Vorono⇢, Georges Vorono¨ ı (1868–1908). Given a set P of points in a space X with distance d, we define the Voronoi cell with kernel p 2 P as the set
V(p) = {x 2 X; d(x,p) 6 d(x,q) for all points q 2 P}.
The Voronoi cells cover the whole space, but they are usually not disjoint.
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Voronoi cells in the plane.
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Tropicalization means, roughly speaking, to replace a sum or integral by a supremum.
Tropicalization means, roughly speaking, to replace a sum or integral by a supremum. So a sum x + y is replaced by the maximum of the two:
x _ y = max(x,y).
Tropicalization means, roughly speaking, to replace a sum or integral by a supremum. So a sum x + y is replaced by the maximum of the two:
x _ y = max(x,y).
Children learn to add numbers, like 5+ 3 = 8 ... but even before that, they learn that 5 comes after 3. So they learn that
5_ 3 = 5 even before they learn that 5+ 3 = 8. Therefore the
max operation is not something new or strange.
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However, an equation like a+ x = b, where a and b are known and x unknown, has, in the set Z of integers, a unique solution
x = b a. The equation a_ x = b sometimes has
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A typical example is the tropicalization of the lp-norm: kxkp = ∑|xj|p1/p is replaced by (sup|xj|p)1/p = sup|xj| = kxk∞. In this case we have convergence: kxkp = ∑|xj|p1/p ! sup
j
|xj| = kxk∞ as
p ! +∞, x 2 Rn.
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More algebraically, tropicalization occurs when we replace addition by the operation of taking the maximum and multiplication by addition.
More algebraically, tropicalization occurs when we replace addition by the operation of taking the maximum and multiplication by addition. Tropical lines and tropical hyperplanes, as well as tropical polynomials are of interest. It seems that there is not yet an axiomatic approach to this kind of geometry.
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We can compare this procedure with the operation of taking the logarithm:
log(xy) = logx + logy, x,y > 0. log(x _ y) < log(x + y) 6 log(x _ y)+ log2, x,y > 0.
We can compare this procedure with the operation of taking the logarithm:
log(xy) = logx + logy, x,y > 0. log(x _ y) < log(x + y) 6 log(x _ y)+ log2, x,y > 0.
So the logarithm transforms multiplication to addition, and a sum to something close to the maximum—the error is small by comparison if x _ y is large.
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A first degree polynomial P(x,y) = ax + by + c is transformed to Ptrop(x,y) = (a+ x)_(b + y)_ c. The zeros of P correspond to the points where two terms of
Ptrop are equal.
A first degree polynomial P(x,y) = ax + by + c is transformed to Ptrop(x,y) = (a+ x)_(b + y)_ c. The zeros of P correspond to the points where two terms of
Ptrop are equal. So this means that we should look at a+ x = b + y, a+ x = c, and b + y = c.
A first degree polynomial P(x,y) = ax + by + c is transformed to Ptrop(x,y) = (a+ x)_(b + y)_ c. The zeros of P correspond to the points where two terms of
Ptrop are equal. So this means that we should look at a+ x = b + y, a+ x = c, and b + y = c. All three are equal at
(x,y) = (c a,c b).
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A tropical straight line in the plane.
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Two tropical straight lines in the plane always intersect.
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Two tropical straight lines in the plane always intersect ... in a unique point ...in a stable unique point ...
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Given two function f and g defined on Rn, we define their convolution product h = f ⇤ g by
h(x) = (f ⇤ g)(x) = ∑
y2Rn
f(x y)g(y), x 2 Rn.
We have to impose some condition to guarantee convergence.
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Let us study the convolution product of two functions of the form ef:
eh1(x) = ∑
y2Rn
ef(xy)eg(y), x 2 Rn,
assuming that f,g are equal to +∞ outside some discrete set. If for instance f, g have their support in Zn and
f(x),g(x) > εkxk C, we have good convergence. The
tropicalization of this convolution product is
eh∞(x) = sup
y2Rn ef(xy)eg(y),
x 2 Rn,
which can be written
h∞(x) = inf
y2Rn(f(x y)+ g(y)),
x 2 Rn.
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Also in this case we have a nice convergence: If we define hp by
ephp(x) = ∑
y2Rn
epf(xy)epg(y), x 2 Rn, p > 0,
then hp converges to h∞ as p ! +∞.
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The function h∞ is the infimal convolution of f and g, denoted by f u g. Here we of course need not assume that f and g are equal to +∞ outside a discrete set. More generally, we define it when f and g take values in [∞,+∞] using upper addition: (f u g)(x) = inf
y2Rn(f(x y)+
· g(y)),
x 2 Rn.
The function ind{0} is a neutral element for u: f uind{0} = f for all f.
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The Fenchel transform of a function f : Rn ! [∞,+∞] is defined as ˜
f(ξ) = sup
x2Rn(ξ· x f(x)),
ξ 2 Rn. Clearly ξ· x f(x) 6 ˜
f(ξ), which can be written as
ξ· x 6 f(x)+ · ˜
f(ξ),
(ξ,x) 2 Rn ⇥Rn, called Fenchel’s inequality. It follows that the second transform ˜ ˜
f satisfies ˜
˜
f 6 f. We have equality here if and only if f
is convex, lower semicontinuous, and takes the value ∞ only if it is ∞ everywhere.
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The Fenchel transformation f 7! ˜
f, named for Werner Fenchel
(1905–1988), is a tropical counterpart of the Fourier
the Laplace transform of a function g: (L g)(ξ) =
Z ∞ g(x)eξxdx.
If we replace the integral by a supremum and take the logarithm, we get
log(Ltrop g)(ξ) = sup
x (logg(x)ξx) =˜
f(ξ), f(x) = logg(x).
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We have (f u g)˜= ˜
f +
· ˜
g 6 ˜ f +
· ˜
g.
We have (f u g)˜= ˜
f +
· ˜
g 6 ˜ f +
· ˜
g.
If ϕ and ψ are convex, then ϕ+ · ψ is convex, but not always ϕ+ · ψ. However, when ϕ = ˜
f and ψ = ˜ g, this is true: ˜ f +
· ˜
g is
always convex, and is often equal to ˜
f +
· ˜
except for a few special cases.
We have (f u g)˜= ˜
f +
· ˜
g 6 ˜ f +
· ˜
g.
If ϕ and ψ are convex, then ϕ+ · ψ is convex, but not always ϕ+ · ψ. However, when ϕ = ˜
f and ψ = ˜ g, this is true: ˜ f +
· ˜
g is
always convex, and is often equal to ˜
f +
· ˜
except for a few special cases. This formula should be compared with (f ⇤ g)ˆ= ˆ
f ˆ g.
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So we discretize by replacing an integral by a sum:
Z
7! ∑, and we tropicalize by replacing an integral by a supremum:
Z
7! sup.
So we discretize by replacing an integral by a sum:
Z
7! ∑, and we tropicalize by replacing an integral by a supremum:
Z
7! sup. There are certain similarities between discretization and
So we discretize by replacing an integral by a sum:
Z
7! ∑, and we tropicalize by replacing an integral by a supremum:
Z
7! sup. There are certain similarities between discretization and
A first step would be to make explicit all similarities and then find a more general procedure, of which both discretization and tropicalization are special cases.
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The Rosetta Stone, 196 BC March 27, found in 1799: Two languages (Egyptian, Greek); three writing systems (Hieroglyphs, Demotic script, Greek alphabet).
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