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This CD-ROM is a collection of virtual experiences presented
at the matetrentino exhibit. Just like the real exhibition, it can
be explored interactively. The CD invites you to confront stimulating
questions about math and gives you hands-on experience in mathematics
or can be used as a guide through a world rich in mathematical facts
presented as a well defined concept.
This presentation aims to highlight an underlying core concept
common to this CD and to the real exhibition and also proposes
itself as a slow-pace guide for a virtual exploration of a selection
of math facts, by explaining the different animations in each section.
 From time to time
each of us looks for the shortest route from point A to point B.
Everyday life is a continuous effort to find the best results with minimum amount of work.
This search is particularly important for science and technology, where often the
mathematical models used to describe phenomena and structures are based on variational
principles, i.e. maximum-minimum problems. In the section maximum and minimum some
of the problems of measuring lengths, areas and volumes of several geometric figures are shown.
If you go Toward the square you see that
among all the rectangles with the same area, the one with the minimum perimeter is in fact the square. If you go Toward the regular polygons, you can make
virtual experiments to extend the concept to all polygons: among all polygons having
the same number of sides and same area, the regular polygon has the minimum perimeter.
What happens if we increase the number of sides infinitely? Going round… you find that
among all the plane figures of a given area the circle has minimum perimeter, or that
among all the plane figures of a given perimeter, the circle has maximum area. This characteristic
of the circle, that a mathematician would call isoperimetrical, helps us understand why many towns have an almost circular shape. This shape, in theory, would give maximum urban
development area and best defensible configuration because of the minimum perimeter exposed to potential
attacks from outside.
If you consider tri-dimensional objects one wonders which shape a solid should take to contain a given volume with
minimum external area, or which figure, having a given external area, has the maximum volume. Toward the sphere helps us deduce the answer.
A second series of animations highlights some questions about minimum length paths on the plane.
To start let us consider Heron's problem:
given two points and a line, what is the shortest path from one point to the other, touching the straight line once?
The solution lets us look at both the light reflection phenomena and the bouncing of a billiard ball problem
in a variational context.
You can then ask yourself which one is the shortest road network connecting a given number of cities.
It might seem strange but finding even The minimum network connecting three points
is not an easy task, you will have to start Hunting for the Steiner Point!
When you have four points, as in the vertices of a square, the problem is even more difficult.
To find the shortest route you have to search in Between X and H.
The last series of animations considers problems similar to the ones above, but in a
higher dimension: which has minimum area among all the surfaces that span a given boundary?
The solutions to this problem belong to a very special family of surfaces that mathematicians
call Minimal Surfaces. Minimal Surfaces are often used to describe the films that form
when you dip a metal frame in soapy water. With a frame made of just two parallel rings the
results are quite complex, the shape of the solution changes dramatically depending on the
distance between the rings. If the rings are close enough to each other the resulting surface
is perhaps the more famous one,
The catenoid.
With different frames you can obtain quite strange surfaces, like
The headphones
and the helicoid. This last one resembling a spiral staircase is apparently very different from the catenoid but it is in fact closely related to it.
Sometimes you might want to ask yourself: is it
Catenoid or helicoid?
Even though it is counterintuitive, soap films often present edges and corners;
these characteristics are called singularities by mathematicians. A description
of this phenomenon can be found in the last two animations of this section, they explore
The singularity of soap films.
Is it possible to start with a 2-d representation (with no further information) and re-build the
3-d environment an image is derived from? What kind of problems will arise when we try to
reconstruct an object starting from just a picture of it? The section visualization is dedicated
to finding answers to these questions and will show, through several experiments, how visuals
are not enough to re-build the initial images.
The first experiment of this section allows the visitor to take a
Flight inside a painting, entering the scene of a possible
three-dimensional reconstruction of the architectural background
represented in a painting by Giovanni Maria Falconetto, now kept in
the Chiesa di S. Maria Maggiore but which originally adorned the organ
of Trento Cathedral.
You can also explore the essential geometries of
this reconstruction by visiting the wireframe version.
In the virtual experiences matetrentino 1 and matetrentino 2,
we have played with the exhibition's logo. At first sight the logo looks flat, drawn on a piece of paper,
but there are endless other tri-dimensional shapes that, when seen from a specific view point, will look
exactly like our logo.
 There are situations in which neither the form nor the dimensions of an object are important nor is the length of an itinerary, but in a sense only more “basic” factors, properties of the objects that would not change even if they were made of rubber and we could twist and distort them as much as we wished (without breaking them). Topology studies these particular properties and the animations in the third section aim to give an idea of what this means in two fields which are apparently very different from each other: one concerning knots and another regarding graphs and surfaces.
With regard to the first field, by making a knot with a cord and then joining the two ends it is “fixed” in something that can no longer be untied (unless you have made an “unknot” which, by manipulating it, you can make a ring).
If we don’t have a physical knot but only a drawing of it, on a computer screen for example, what we see is a curve which twists and intersects in certain points: in these points the drawing allows us to understand how it would be possible to physically produce the knot using a cord, by indicating which part passes over and which passes under. In Crossing and colours: draw your own knot you can start to “familiarise” yourself with knots: you can play around and simply try to make “pretty shapes”, or you can try to “control” what you are doing and understand which moves are needed to create certain design.
If we talk about knots in Trento, we can’t help remembering the magnificent knot that connects the four columns of the Cathedral’s transept. Nest of knots gives us the opportunity to “capture” this knot “by closing ” the columns. But here, differently from what happens with knots made on a single cord which can be fixed in a unique way by joining the two ends, instead there are many possibilities and we can capture knots which are very different from each other.
A knot also appears on the coat of arms of an old family of Trento. It shows a lion with its tail twisted into the shape of a love-knot (or Savoy knot) which, once closed, mathematicians call a figure-eight knot. However the knot is not always shown correctly in the coat of arms: sometimes one part that passes over is mistaken for one that passes under… and this is enough to produce completely different knots, which you can try in the animation Knot or unknot?
Another animation, Borromean rings, shows a possible, perhaps unexpected, genesis of Borromean rings, a three-component knot with a bizarre characteristic: if we consider only two of these components they are untied while no one part can be slipped off of the block of three.
In the animation Two numbers for a knot we can discover a particular family of knots (torus knots) which it is possible to describe completely using two integers numbers. The family also includes some simple knots (such as, for example, the trefoil knot, the two-component knot made by two rings connected in a chain and the so-called Solomon’s knot) which can be created by playing with this animation.
The central theme of the second field draws on a classic problem from graph theory, known as the problem of the three houses. In
Paths without crossings
you are asked to connect three points (which could represent three houses) with three other points, tracing paths that do not intersect each other. This problem is given in four versions: the first reproduces the “normal” situation of the drawing on a sheet of paper, while the other versions allow us to leave the sheet on one side to enter again from the opposite side while observing precise and fixed indications (for example, re-entering at the same height). On the right side of the animation we can see how imposing these “rules” corresponds to tackling the problem no longer on the plane but on different surfaces (the surface of a torus, a cylinder or a Möbius band). In reality not all these problems are solvable: the problem has no solution on the plane and on the cylinder and the reason essentially lies in the fact that on the plane and on the cylinder (and also on the sphere) any curve which is closed (i.e. which returns to the starting point) and simple (i.e. it has no self-intersections) divides the surface into two parts. The animation Cutting a surface illustrates this fact and how this is not true in the case of the torus and Möbius band, on which the problem of the three houses actually has a solution.
The last animation From the rectangle to… shows us why the rules established in Paths without crossings correspond to tackling the problem on a surface other than the plane.
 Symmetry is a basic interpretation often used (in a more or less conscious way) to understand the most varied messages coming from the surrounding world. The fourth section of the CD is dedicated to symmetry (and asymmetry), the animation helps us identify the different kinds of symmetry.
The first virtual experiment allows us to visit an Image Gallery.
By visiting this section one begins to understand the problems of classifying figures with respect to their symmetry type.
In the image gallery, each row contains three images. The symbol on the left of each image denotes the symmetry
shared by the diagrams. By clicking on one of these images, you are provided with an enlargement in context.
By clicking on the group symbol you will get a short animated explanation of the structure of the group.
Some of the hypertext pages illustrate various possibilities:
Rosettes:
where there is no translation keeping the figure unchanged; Friezes:
where there exists a translation, and every other translation fixing the figure can be obtained by
iterating a given basic translation; the types of symmetry possible in this kind of imagery are exactly
seven and they differ according to which other transformations may fix the image (besides the basic
translation and its iterations) and finally Wallpaper patterns:
images where there are many translations, in different directions, fixing the picture and in this case,
the possible symmetry types are exactly 17.
You can try Draw your own frieze.
This interactive applet lets you choose one out of seven different types of symmetry
(i.e. one of seven possibly different ways of repeating the image) and then position
some shapes in a specific area of the screen. You will thus obtain a frieze that replicates
the selected shapes according to the selected symmetry. Among the various shapes available,
there is also a foot. You can then try to reconstruct seven different ways of walking in order
to obtain the seven distinct friezes as if they were footprints.
Similarly, you can Draw your own wallpaper pattern.
You will use a virtual mirror box, from which you can choose various shapes. By putting some of the available
polygons in the mirror box you get a wallpaper pattern … although one should note that not all the 17 patterns
can be obtained in this way.
Finally, the user is asked to Recognize a frieze.
You will be asked to compare photographs, that appear on the screen, to the seven different patterns
of a frieze and to understand which one is the pattern in the image. In order to do this, you will click
on the available buttons to see the effect of the corresponding transformations on the photograph. You can
understand which are the ones that fix the picture and these are the ones that characterize the symmetry pattern.
Similarly, you can try to Recognize a wallpaper pattern.
The photograph on the screen is the image of a wallpaper pattern. You will have to identify its symmetry pattern out of the possible 17 presented there.
Suggestions will be given when you supply an incorrect answer and they will help you in your following attempt.
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