The Archimedes Mirror
Hans Grassmann, Isomorph srl, Univ. of Udine
La fisica communica con Medicina, Arte, Spiritualita'
ICTP Trieste, Nov. 2014
1000 b.C.
Genesis
Anaximander
Anaxagoras
Democrit
Pythagoras
2000
0
Death of Archimedes
Aristotele
Luther
Kopernikus
Brahe
Kepler
...others
Newton
Huygens
Galilei
Physics
One example of “physics disappeared”:
Mirror of Archimedes
According to Tzetzes, Galeno, Cassio Dione Cocceiano, Giovanni Zonara,
the mirror of Archimedes did exist.
In modern times several “tests” were made to verify these accounts,
One test on request of the American President, B. Obama.
Those “tests” failed to ignite wood, in the Obama test an array of hand held
mirrors achieved only 100 °C.
it was concluded, that the burning mirror of Archimedes had not really existed.
Objection:
the mirrors used in these tests correspond to the present state of the art of light
concentration, but do not correspond to the historical record and they are not
compatible with the laws of physics.
http://en.wikipedia.org/wiki/Burning_glass
http://it.wikipedia.org/wiki/Specchio_ustorio
Physics objections
Parabolic mirrors have their focus between
the mirror and the sun,
If one reflects light sidewards,
the focal point gets lost.
=> Parabolic mirrors cannot burn ships
Arrays of plane mirrors (as used in modern
solar tower) need a computer for their
control.
If one adjusts the mirrors by hand, one
cannot see by eye where each individual
mirror is reflecting the light (due to the
strong illumination of the focal region).
=> arrays of hand held mirrors cannot burn
ships
In order to make an array of mirrors dangerous
for a ship (without computer control)
the mirrors must be connected mechanically,
so that they can be adjusted concurrently.
In state of the art systems, this is not possible.
The Antikythera Mechanism
The burning mirror of Archimedes is not his only achievement, which got
lost. According to literature, Archimedes also built a very high quality
astronomic clock,
« Nam cum Archimedes lunae solis quinque errantium motus in sphaeram inligavit,
effecit idem quod ille, qui in Timaeo mundum aedificavit, Platonis deus, ut tarditate et
celeritate dissimillimos motus una regeret conversio. quod si in hoc mundo fieri sine
deo non potest, ne in sphaera quidem eosdem motus Archimedes sine divino ingenio
potuisset imitari. »
Cicerone, Tusculanae disputationes
« In realtà, quando Archimede racchiuse in una sfera i movimenti della luna, del sole e
dei cinque pianeti, fece lo stesso che colui che nel Timeo edificò l'universo, il dio di
Platone, e cioè che un' unica rivoluzione regolasse movimenti molto diversi per
lentezza e velocità. E se questo non può avvenire nel nostro universo senza la
divinità, neanche nella sfera Archimede avrebbe potuto imitare i medesimi movimenti
senza un'intelligenza divina. »
again, modern scientists did not wish to believe the literary account:
Mechanism of Antikythera
Even when in 1900 one of
the astronomic clocks
described by Cicero was
found in a roman
shipwreck near the island
of Antikythera.
Antikythera
Ephebe
scientists just refused to believe that it
could be real and ignored it for 51 years,
until the studies of de Solla Price.
National Archaeological Museum
of Athens
Michael Edmunds, who led a 2006 study of the mechanism, described the device
as "just extraordinary, the only thing of its kind", and said that its astronomy was
"exactly right". He regarded the Antikythera mechanism as "more valuable than
the Mona Lisa
http://en.wikipedia.org/wiki/Antikythera_mechanism
For the following discussion note:
the Antikythera Mechanism
is linear,
while the planet movements
seem to be complex
From a physics point of view a mirror system and
an astronomical clock are similar :
When imagining in your mind the mirror system,
do not use the mirror system as a reference,
rather observe the mirrors from the focal point:
Watching the N mirrors from the focal point it
seems, that there are N suns in the sky, and each
mirror is following the position of one of them
This is identical to the task of an astronomic
clock, which shows the position of several
celestial bodies in the sky.
=> who could build the Antikythera Mechanism,
could also build the burning mirrors
A system with concurrently moving mirrors, corresponding to the description
of John Tzetzes does exist,
John Tzteztes (twelfth century AD):
>> Archimedes … set similar small
mirrors with four edges, moved by links
and by a form of hinge <<
The system was developed by Isomorph srl at Trieste at the beginning of the 21th
century AD
GREEK MATHEMATICAL WORKS, Tranlated by Ivor Thomas, Loeb Classical
Library, Harvard Universiyt Press, Cambridge, 1941, Vol 2, page 19
10
The system is called
Linear Mirror II
9 kW thermal energy
With water: 100 °C
(also in winter)
With oil:
(also in winter)
180 °C
Can substitute 700 to 800 liters of oil per year
Ready industrial product with
European Solar Keymark certificate,
eligible for state incentives:
Isomorph srl and
Isomorph Production srl,
www.isomorph-production.it
Functioning of Linear Mirror II
For details see www.isomorph-production.it
In the present version of the Linear Mirror, the
focal region is at a fixed position for
convenience – the possibility to keep the focal
region fixed is one important advantage of the
system.
If instead one wants to follow the movement
of a ship in direction and distance – for
example – the focal region can be made
adjustable as well.
Hotel “Al Cavliere”, Pordenone
Compared to conventional plants:
Simpler
Cheaper
less energy loss
Less specific heat
Intelligent
cradle to cradle
complete test report public on
www.isomorph-production.it
13
The Linear Mirror II can also perform solar pyrolysis:
transform simple biomass (straw, hay, waste) to carbon and gas
Solar carbon from straw:
Heating value : 27.2 MJ/kg
Solar carbon and solar gas
- CO2 neutral
- compatible with existing
infrastructure/technologies
- store solar energy for long
periods of time without loss
Solar gas
14
art
The Evolution of art until Renaissance went together with an always improved
resemblance of depiction :
However, the achieved capacity of
resemblance was used to depict scenes,
which do not exist, and therefore cannot
have any resemblance whatsoever :
In the following centuries art seems to have searched for what it
really wanted to achieve: impressionism? expressionism? cubism?
The crisis of European culture at the beginning of the 20th century shows the
limits of traditional art
If humans would
have taken paintings
like these seriously
the horrors of the 20th
century never could have
happened
This leads to a new era of art, as for example explained in the works and words
of Joseph Beuys - “erweiterter Kunstbegriff”.- extended concept of art
and correspondingly to a new subject of art – social sculpture.
In 2012 the artist Elena Mazzi started her project “Reflecting Venice”.
supported by the foundation Bevilacqua la Masa at Venice
http://www.bevilacqualamasa.it/atelier-elena-mazzi
Elena uses a Linear Mirror as an interface,
connecting local reality,
history, and
options for the future.
For Elena's project, the
reflecting surfaces of a Linear
Mirror II system are made from
venetian class instead of
aluminum.
An artisan at Murano engraves these mirrors in the historical venetian technique
with pictures of plants, which are in decline in the lagoon of Venice
A Linear Mirror, equipped with these
engraved glass mirrors was to be
installed at some commercial
location at Venice, providing energy
for the contemporary world, like for
a hotel, or some company, or a
restaurant. This location could
furthermore have been a connection
to the future, since the production of
the Linear Mirror itself would be very
well suited for Venice: the
production of the Linear Mirror is a
quiet industry, without much energy
consumption, noise, pollution or other hazards, it could operate well in Venice, as
a new branch of the dying historical glass industry.
now a strange thing happened: The Linear Mirror with its engravings and
references was supposed to passively reflect the past and present, and also
some future opportunities.
But maybe because it did
this so well and so real, the
passive process of
reflection became an active
one. The mirror began to
reflect things, which it was
not supposed to reflect,
making these things
happen on its own device,
as if it were alive:
It turned out, that nobody at Venice wanted to have Elena's mirror, not even for
free. Elena tried for more than one year to convince some company or hotel or
restaurant or association to install a free mirror for producing 800 Euros of
income each year. But nobody wanted that.
So the mirror did not only passively reflect on the decline of Venice, but
interacting with its social environment it uncovered, made explicit and explained
the reason for this decay.
Reflecting Venice
was bought by Botkyrka Konsthall
(Stockholm) and shown at the
Biennale for Architecture at Venice
as part of the Fittja Pavillion.
The Fittja Pavillion was voted to be one of the most
interesting ones at the Biennale by the journal Domus.
There are many other projects, which show how important the
connection between art and physics is today, for example
Fabiola Faidiga, Contemporary shower
Work in progress – we are looking for sponsors (could be you ?)
There is a not so small problem:
Many people are not able anymore to
understand art. They wonder:
What is art?
What does an artist do?
As history shows, the flourishing and
perishing of art and physics are
correlated.
So physics should worry about this
problem.
And in the same way – as physics should worry about art –
society as a whole should worry about physics:
Had the Archimedes mirror not been lost 2.300 years ago,
the “carbon era” could have been avoided,
avoiding global heating, a number of wars and so called “cultural conflicts”.
The total absence of research on topics like this is frightening:
How could it happen, that this important technology was lost?
How could It remain in oblivion for 2.300 years?
Why do we spent billions each year for solar energy, without any technical
progress in the real world of solar energy?
If somebody would re-invent the Archimedes mirror today, how would society
react?
From this perspective, and since above questions go beyond the scope of this
presentation, we do just a first little step towards these questions by asking
what artists are doing, from a physics point of view?
Art & physics
What does art from a physics point of view?
(Prolegomena)
Each painting can be represented by a 2 dimensional pixel array
to any desired precision there exist MN different arrays, forming a list of MN elements (“big list”)
(N=number of pixels, M=number of colors and intensities)
most arrays show “white” noise.
fraction of arrays showing a possible image of natural world is small beyond
imagination <=> “reality is extremely unlikely” (unplug a TV set and wait for a
movie to happen by chance)
Human beings are able to identify those rare images.
Artists identify the subsample of “important” or “relevant” images,
they communicate those images to other human beings by pointing to the
image - creating relevance.
These considerations can be extended to 3d works of art (statue), and temporal
sequences (movies, performances)
Some of the elements of
the list of arrays
cannot be found in the
material world,
a painting needs to be
created, in order to perform
the act of pointing.
Some elements of the big list can be found in nature or can be arranged.
The artist can point to them by means of a photograph or a performance,
without the need to produce a painting.
Painting is a form of pointing.
There is no preferred form of pointing,
relevant is only the relevance found
Interesting physics questions arise with regards to art:
How many natural images are there?
How many relevant images ?
Can one explain or derive these numbers?
How much information is contained in the Universe?
How can the brain identify these subsamples, given the small number of
10
10.000.000.000
(mirror-) neurons? (« 10 compared to O(10
) arrays)
Question to be asked in order to obtain a Linear Mirror system:
How many motors are needed to make N mirrors follow the sun?
(N mirror problem)
The key point of this presentation is the following observation:
the green (art) and the red (Linear Mirror) questions are very similar
both require an understanding
of what one might call “physics of information”.
Attempts to discuss questions of this kind in terms of physics do
exist. For example:
:Phys.Rev.Lett.88:237901,2002:
Computational capacity of the universe, Seth Lloyd
>> Merely by existing, all physical systems register information. And by evolving
dynamically in time, they transform and process that information. The laws of
physics determine the amount of information that a physical system can register
(number of bits) and the number of elementary logic operations that a system
can perform (number of ops). The universe is a physical system. This paper
quantifies the amount of information that the universe can register and the
number of elementary operations that it can have performed over its history. The
universe can have performed no more than 10120 ops on 1090 bits. >>
But these attempts are usually based on Shannon's framework, where
messages (and paintings) have no meaning, and where the amount of
information is measured without defining what information is.
Neither art nor physics exist in that world, not even a Linear Mirror II.
From a physics point of view, the subject of information processing are
“messages”, for example digital messages.
And from a mathematics point of view these (here digital) messages are vectors
of the arithmetic vector space, F2n.
Information processing can therefore be expressed in terms of a vector
transformation T, transforming an input vector a to an output vector b
b = T(a)
Note, that the number of microstates of the Universe is final (Boltzmann) and
that therefore finite message are sufficient for all physical messages.
This is different from mathematics or information theory, where one of the
central problems is the halting problem, associated to the possibility of infinte
messages.
Considering b = T(a)
“information” can be defined by its conservation:
If an information processing system T transforms a message a to a
message b, b = T(a) and if there is a = T-1 (b)
then T conserves information, and
a and b are identical messages – have the same meaning
Of course, identical messages must have the same “amount of information”,
and since a and b will in general not have the same number of bits, the length
of the message cannot be a measure for the information content.
Already here we are incompatible with Lloyd and Shannon
(In our new scenario the amount of information might be measured by the
number of dimensions of the message space.)
In the following we consider only injective transformations, to shorten the
discussion, while non-injective transformations are very interesting, too.
In the simplest case, T is linear. It is then true that
T maps Input vector space U on isomorphic output vector spave V
T is a matrix, consisting of vectors, which again form a vector space
isomophic to U and V.
in everydays language: input and output messages may look different, but are
not, and also the information processing system is not different from the
messages. The information processing system must contain the same
information as the input messages, before it can operate on them.
Example: y=2·x
input
Digital numbers
No algorithm,
no energy dissipation,
no fundamental operations
T
0001
00001
0010
00010
0100
00100
1000
01000
10000
output
For a non-linear T the situation is not much different:
we can always project the 2n vectors of U and V to 2n base vectors
so the operation T in the digital space can be substituted by a new
operation D operating between two bases – D can therefore be written as
a Matrix.
operation b=a on
canonical numbers.
2
000001
000001
000010
000010
000100
001000
100000000
Lookup table
this is a RAM (random access memory)
If not tied to a digital space through a multiplexer,
It can perform calculations concurrently
It therefore emulates a quantum computer – without disentanglement problem
Another example:
images may be created by
algorithms performed by computers,
like in a digital camera.
But pictures can also be taken by a
pinhole camera,
and as for example the work of the
artist Luigi Tolotti shows, pinhole
pictures are of high quality and
relevance
Here, the algorithmic Turing machine
is substituted by a hole.
And of course, also the DNA and the human brain are organized as lookup
tables (for instance “mirror neurons”),
they are not good at performing algorithms.
And just because of this, they are extremely powerful.
A wide field of research opens here!
Algorithms are of course allowed, and they are defined in our scenario:
Algorithms and algorithmic machines are ordered systems of lookup tables.
Note. That also in a Turing machine the use of lookup table cannot be avoided,
since the fundamental operations cannot be grounded in algorithms.
a1
a2
b1
+
Advantage: small lookup tables
Price to pay: loss of concurrency
At the time of Turing, large lookup tables
were not available, today they cost 3 €
b2
+
+
+
c1
c2
c3
1
1
1
Complexity:
number of lookup table elements of algorithm compared to number of full
(concurrent) lookup table.
Algorithms are defined and allowed for in our scenario,
But they can as well be avoided
In the Shannon world, the position of each mirror of a N mirror system
needs to be evaluated in two dimensions by an algorithm of its own, and
therefore it must have 2 motors of its own correspondingly.
In our scenario this need for 2N algorithms does not exist anymore, one
motor should be enough, it is needed for generating input number, it is a
real time clock.
Some of the statements made refer to the vector space F2n and are true only
with respect to this.
Is it, for example, true, that also a T, which processes images (creating a correct
description of the image) is an isomorphism?
For discussing pictures, Boolean Algebra is well suited
We find:
1) m physics objects form a 2m – element Boolean algebra
(that is, how Boolean algebra is taught at school: take objects like an apple,
an orange, a car, and form ensembles with them: you can group them in 2m
different ways)
(In this context, objects at different positions are considered different objects.)
2) also the m images of the m objects from a 2m –element Boolean
Algebra
It is then possible to
associate to each image ai its correct description bi, bi = D(ai).
And it is by construction true that
the image of a photograph of two objects will be the same as the logical-or of
two images, each one showing one of these same objects:
D(ai V aj) = bi ∪ bj.
D is then an homomorphism,
and since D is bijective, it is also an isomorphism.
Consequence:
in the list of all arrays describing all possible pictures
we do not need 2m elements in order to describe the 2m different
images of m objects, but only m elements.
From a practical point of view:
- Create the largest technically possible complete list of all images possible, for
instance these may be images of 5x5 pixels.
Use them as Boolean Atoms.
- Create total orders from these atoms (lines, curves etc)
(This is done only by Boolean operations, no probability, do decision making.)
- continue this process in a linear way.
- occlusions and reconstruction problems are handled by Boolean Ideals.
>> an atom in a Boolean Algebra is a nonzero element a such that there is no
element b with 0 < b < a
Here a ≤ b precisely when a = a Λ b <<
>> a total order, linear order, simple order, or (non-strict) ordering on a set X is any
binary relation on X that is antisymmetric, transitive, and total. <<
>> an ideal of S is a subset J such that for all a, b in J we have
(a V b) in J and for all x in S we have (a Λ x) in J. <<
For a more detailed discussion see Isomorph Letters:
http://www.isomorph.it/letters/articles/on-the-mathematical-structure-ofmessages-and-message-processing-systems
Sine in the context of this presentation a detailed discussion of Isomorph image
identification is not possible, we show instead an example,
written along the lines described in the previous slide:
No probabilities, no
decision trees, no fits
Linear structure – if we
want to identify
elephants, it is sufficient
to add one subroutine.
.This program is executed on a Turing machine. Otherwise it would give a
concurrent description of what it sees.
We have shown
that Grassmann Algebra can be used to discuss the particular case of
“algorithmic information processing” and
Boolean Algebra for the particular case of understanding the meaning of
pictures
Conjecture:
Physics of information should in more general be based on category theory
also the human brain should be an “isomorphy” (ideally) of the outside world, or
a mirror of the world (“was der Fall ist”, Wittgenstein)
In full agreement with the findings of Rizzolatti, Kant et al.
But where does it come from? Where does the a priori come from?
While information processing always can be done without energy dissipation
It remains true (Boltzmann), that the creation of the information processing
system (T) must dissipate energy.
At which point the second law of Thermodynamics applies:
The formation of brains becomes a spontaneous process,
as a live form of its own with its own evolution (ordering “bits”)
like organic live is a live form of its own (ordering molecules)
Note however, that information has now become “relative”, a brain cannot have
an order by itself, it cannot create its own reality, it can only be a mirror of
reality,
Or an image, of what there is (Wittgenstein)
As a consequence, human consciousness can understood as the
consciousness of the world,
which indeed would not be conscious of its existence without the human.
It is often claimed, that science has “marginalized” the human being
(Kopernikus, Darwin, Freud, genetics)
From what was said in this presentation it becomes clear, that this
“marginalization” was based on a misunderstanding:
Up to now, science has discussed the human only as a form of organic live.
With respect to organic live, the human being is indeed marginal.
With respect to intellectual live, the human being is moved back to the “center
of the world” and is different from the animals.
Conclusion
Art is therefore (like physics) a live form of its own, a form of intellectual live
Its evolution is tantamount to improving our ability to see the world and
to be the world, and as its living mirror to be the consciousness of the world.
Searching for relevance, art makes the human relevant.
Like all live forms, art (like physics) is spontaneous, but it is not a necessity
– it can be extinguished
2000
0
1000
Genesis
b.C.
Anaximander Death of
Luther
Newton
Anaxagoras
Kopernikus
We should not do that,
Archimedes
Huygens
DemocritAristotele
Brahe
Galilei
we should not extinguish it
Pythagoras
Kepler
Because we would cease to be images.
We would drift away from the center of the world
and loose ourselves.
Maybe we should study category theory next?