Creating Dynamic Geometry Constructions As Composition Tools in Art and Photography Modelling Optical Lenses with Dynamic Geomet

CreatingModelling dynamic optical lenses geometry with constructions Dynamic Geometry as composition Software tools in art and photography Andreas Ulovec Evgenia Sendova, Toni Chehlarova 1 Introduction Institute of Mathematics and Informatics, Bulgarian Academy of Sciences In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way I.of making Introduction light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustmentsSeeing to the system is not canas simpleusually asonly it looksbe done by removing one piece and putting another piece in. To see what happens if you Admake Reinhardt a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them Manyobserve artists how theclaim path that of light they actually could changes.explain nothingWe want aboutto demonstrate their works, how you that can the showir paintings the path cameof uponlight through them bya lens inspiration. with the help The of dynamic founder geometry of the software abstract (DGS). art however, Vassily Kandinsky, expresses in the book ―Concerning the spiritual in art‖ his theory of painting and sums up This material can be useful for science teachers, who can use it to model experiments with lenses, ideas that influenced his contemporaries. He makes the brave prediction that we are fast reflection and refraction – not instead of the actual experiment (if one sees experiments only in approachingsimulation, the the pedagogic time of valuereasoned is not andquite conscious the same), composition,but complementing when it. the It canpainter as well will be beuseful proud tofor declare mathematics his work teachers. constructive Well, now. where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and Toanother motivate part penetratesbetter the the study glass of and geometry continues for there, students in another with angle.interests The in same art wehappens could when reveal the for themlight reaches the strong the other relation surface between of the thelens esthetics– again mathematics of artistic is compositions required to calculate and some the angle geometric in principles.which the light When is ref readinglected and the refracted. works of For art ideal critics lenses, we there come is anacross easy equationnotions calculatingsuch as harmony, these style,effects rhythm, – but this balance is just a(not model, necessarily and it does the work better well defined only with rules, thin lensessymmetry, and with geometry light falling). Perhaps in theynear thinkthe centre that ofif revealedthe lens. Withthe ―rules‖ thicker behindlenses anda balance light being composition more off- centre,would thetrivialize calculations the art. become more complex, and from the equations alone it would be difficult to see what happens. With ToDGS us, it revealingis possible certainto simulate patterns the properties and rules of awould lens without in contrast actually raise having the to level use aof lens, appreciation laser light, of anetc. observer. But even forThe the modern DGS, we fine need art mathematics tries to speak to create about the things simulation which in willthe first be place.seen, that is why its language is not understandable for many. But this language could be better learned if we try to study2 Easy it together beginnings with the language– light hitsof geometry. a plane surface Let us consider several relatively simple geometric constructions which have proved useful in creating2.1 Reflection and studying the balance of the fine-art compositions. After describing them we shall implementWhen a ray themof light in hitsa dynamic a plane glasssoftware surface, environment a part of it (GeoGebrais reflected. T inhe our law case) of reflection so as to says illustrate that howthe angle they ofcould incidence be applied (between to theexploring ray of light various and the paintings normal) is(classical equal to the and angle more of modernreflection: alike).

II. Rabatment

A compositional method broadly used in the 19th century was the rabatment. This method consists of taking the shorter side of a rectangle and placing it against the longer side (rotating the shorter side along the corner), creating points along the edge that can be connected directly across the canvas as well as a diagonal from these points to the corners. In a rectangle whose longer side is horizontal, there is one implied square for the left side and one for the right; for a rectangle with a vertical longer side, there are upper and lower squares. In traditions in which people read left to right, the attention is mainly focused inside the left- hand rabatment, or on the line it forms at the right-hand side of the image (Fig. 1)

Fig.1 Reflection of light at a plane surface.

This project has been funded with support from the European Commission in its Lifelong Learning Programme (510028-LLP-1-2010-1-IT-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you haveFig. to 1 take The out left the rabatment current lens and and its put appearance in the new in one. a Monet’s Students paintingcan then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them Toobserve achieve how a the more path powerful of light actually composition changes. one We couldwant to add demonstrate the diagonals how you of thecan showrectangle the path and of the light through a lens with the help of dynamic geometry software (DGS). two squares. The focus of attention is clearly seen in the painting of Giotto in Fig. 2 This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

2 Easy beginnings – light hits a plane surface

2.1 Reflection When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection:

Fig. 2 The rabatment as applied in paintings by Giotto and Bogdanov-Belsky

Let us now create a dynamicFig.1 rabatment Reflection in ofGeoGebra light at a plane as a specialisedsurface. button of its toolbar,

III. Creating a Rabatment button in GeoGebra Modelling optical lenses with Dynamic Geometry Software We shall start with constructing a dynamic rectangle. This could be done in multiple ways but weAndreas shall Ulovecuse two of them which turn out to be the most appropriate for creating a Rabatment button. First1 Introduction approach – we construct as independent objects the corner of the rectangle (point A) In optics when it comes down to show the path of rays of light through glass, lenses or systems of andlenses, two many sliders physics for its teachers width groana and – length the experiments b by means are ofquite the complexbutton , and you and need specifying a lot of the intervalequipment. to beIt is [1, difficult 10]. enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke Weglass define lenses point etc. Now B (the that’s opposite not always corner available, of the rectangle)and adjustments by entering to the system in the can command usually only row: be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place. Fig. 3 Constructing two opposite corners of a rectangle 2 Easy beginnings – light hits a plane surface Next we construct lines through A and B parallel to the coordinate axes by first showing the 2.1 Reflection coordinate system: When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection:

and then using the button for a parallel line:

Let us remind that to do this we first click on the point and then on the axis parallel to the line under construction. The remaining two vertices of the rectangle could be obtained as intersection points of these lines, i.e. by clicking on theFig.1 button Reflection andof light then at consecutivelya plane surface. on the lines.

ThenModelling we construct optical a quadrilateral lenses by means with of Dynamic the button Geometry Software with vertices the fourAndreas points. Ulovec Thus we constructed an object (polygon1) with sides the segments a1 , b1 , c1 , d1 . 1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the Letlight ba reaches. Now the otherwe construct surface of circles the lens with – againcenters mathematics the four verticesis required of theto calculate rectangle the and angle a in radiuswhich theb – light the islength reflected of the and shorte refracted.r side For of ideal the rectangle.lenses, there We is anuse easy for equationthe purpose calculating the button these foreffects a circle – but by this a centeris just a through model, and a point it does (t hework point well being only withthe other thin lenses end of and the with segment light falling b): in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

We2 Easy find the beginnings intersection points – light of the hits four a circlesplane with surface a side of the rectangle and construct two2.1 ofReflection the rabatment segments. Then we complete the construction with the diagonals. When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection:

What is left is to construct the square in the center which appears when b a2 b . (If necessary, we move the slider and construct the intersection points needed and the square.)

Fig.1 Reflection of light at a plane surface.

Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in Wesimulation, hide the the auxiliary pedagogic objects value is(the not lines quite and the same),the circles) but complementing and we explore it. It the can construction as well be useful for variousfor mathematics values ofteachers. a and Well, b . now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place. It is clear that if ab only part of this construction remains. We make similar constructions so2 asEasy to make beginnings a workable construction– light hits for a a plane rectangle surface with a shorter base. 2.1 Reflection When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection:

After hiding the auxiliary objects we get a construction ready for explorations. To make the things easier for implementation we shall enrich the tool bar with a Rabatment button. Here it is how we can create a new tool:

Fig.1 Reflection of light at a plane surface.

Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction WeIn optics select when the objects it comes of down the rabatment to show the construction path of rays (segmentsof light through and polygons):glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS).

This material can be useful for science teachers, who can use it to model experiments with lenses, Thereflection inputs and are refractioninserted automatically– not instead of(a thepoint actual and experimenttwo numbers (if inone our sees case): experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place. We name the button (later we could make a suitable icon): 2 Easy beginnings – light hits a plane surface 2.1 Reflection When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection:

The button has been successfully created. After clicking on it we have to enter a point, a number (we shall use the slider a ), and another number (the slider b ).

Fig.1 Reflection of light at a plane surface.

This project has been funded with support from the European Commission in its Lifelong Learning Programme Then(510028 the-LLP rabatment-1-2010-1- constructionIT-COMENIUS appears-CMP). This on publicationthe screen. reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Second approach – two points, the ends of the diagonal of the rectangle, are constructed as independentModelling objects. optical (If we lenseswould like withto explore Dynamic the ratio of Geometry some segments, Software we should additionallyAndreas Ulovec display their lengths.) Then we continue similarly to the First approach and we construct a rectangle in which the points A and B are free (movable) points. 1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in Itwhich is convenient the light is to ref constructlected and two refracted. Rabatment For ideal buttons lenses, in there the sameis an easyGG equationfile (named calculating these RabatmanPNNeffects – but this andis just RabatmanPP a model, and after it does the work necessary well only inputs with forthin the lenses respective and with construction light falling in – a pointnear theand centre two numbers of the lens. in the With first thicker case, lensesand two and points light –being in the m oresecond off-centre,). the calculations become more complex, and from the equations alone it would be difficult to see what happens. With ThenDGS itwe is possibledelete the to simulate constructions. the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place. In addition, we construct two sliders for the parameters to be used with the Rabatman PNN button,2 Easy and beginnings two points to position– light the hits images a plane. surface 2.1 Reflection When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection:

IV. Inserting images on the GeoGebra screen In order to explore a specific painting (saved preliminary in a graphic file) we shall first display it on the GeoGebra screen by means of the Insert Image button:

Fig.1 Reflection of light at a plane surface.

Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With ToDGS be it ableis possible to resize to simulate the insertedthe properties image of a by lens preserving without actually its proportions having to use ita lens, is convenient laser light, to associateetc. But even two for of the its DGS, vertices we need with mathematics two preliminary to create constructedthe simulation points, in the first e.g. place. A (a free object) and B – lying on a horizontal line through (parallel to the X axis). 2 Easy beginnings – light hits a plane surface 2.1 Reflection When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection:

Then we can use the Rabatment button, place the rabatment construction on the image and look for interesting properties of the composition. Here are some experiments with works by one of the most talented 20th century Bulgarian artists Vladimir Dimitrov – the Master.

Fig.1 Reflection of light at a plane surface.

Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

Now you might want to exploreFig.1 Reflection the rabatment of light at compositiona plane surface. method in the works of such famous artists as Delacroix, Ingres, David, Degas, Sargent, Henri, Cassatt. This project has been funded with support from the European Commission in its Lifelong Learning Programme (510028-LLP-1-2010-1-IT-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

V. Some simpler geometric constructions for exploring art Modelling optical lenses with Dynamic Geometry Software composition Andreas Ulovec a. The central rhombus The1 Introduction main idea (the logical emphasis) of a painting is often located in a rhombus with vertices the midpoints of the sides of the rectangle: In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place. Task 1. Construct a rhombus button in GeoGebra for exploring images. Hint2 Easy. Start beginnings with the construction – light of hits a rectangle a plane similarly surface to the Rabatment construction. 2.1b. ReflectionThe rule of thirds When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that Thethe anglerule ofof incidencethirds is (between a simple the method ray of light that and can the be normal used) isnot equal only to theas anglea tool of for reflection: exploring the paintings of famous artists but also to enhance and improve our own compositions (when we draw or take pictures).

In the diagram below, a rectangle has been divided horizontally and vertically by four lines. The rule of thirds states that the points of interest for any rectangle is determined by those lines. The intersections of the lines are considered by some specialists to be power points (the black dots in Fig. 3).

Fig.1 Reflection of light at a plane surface.

This project has been funded withFig. suppo 3 Thert from rule the of Europeanthirds and Commission the power in its points Lifelong Learning Programme (510028-LLP-1-2010-1-IT-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Here is the rule of thirds in action (in horisontal version): Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place. And in vertical one: 2 Easy beginnings – light hits a plane surface 2.1 Reflection When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection:

In both paintings above the woman’s head is located within the central third of the rectangle but the positions of the hands define different orientation of the frames. When applying simultaneously the vertical and the horizontal lines we get a central square which often plays a special role. Here are examples of two French impressionists:

Fig.1 Reflection of light at a plane surface.

Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them Theobserve rule how of thirds the path could of light be actuallyapplied changes. easily when We want making to demonstrate photographs how youof a can scenery show the- putting path of the horizonlight through 1/3 a of lens the with way the fromhelp of the dynamic top orgeometry 1/3 of software the way (DGS). from the bottom creates a more attractiveThis material composition. can be useful Similarly, for science this teachers, rule works who withcan use vertical it to model elements experiments instead with of horizontal lenses, ones.reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

Task 2 Make several digital pictures of a scenery by applying the rule of thirds in just one of them. Explain which version seems the most balanced one.

Fig.1 Reflection of light at a plane surface.

Task 3 Create Thirds buttons (a vertical and a horizontal versions). Modelling optical lenses with Dynamic Geometry Software TaskAndreas 4 UlovecExplore some classical and some modern paintings by all the composition buttons. Hint.1 Introduction Insert images from the virtual galleries on the Internet following the instructions in Section IV. Use consecutively the different composition buttons, e.g. In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

VI. The golden section in art The most famous mathematical compositional tool, though, is the Golden Ratio, also known as the Golden Mean or the Golden Section. It is defined as the point at which a segment can be divided in two parts a and b, so that the ratio of the longer part (a) to the whole (a+b) is the same as that of the shorter part (b) to the longer one, i.e. a/a+b = b/a:

Fig.1 Reflection of light at a plane surface.

This ratio is denoted by Φ. A rectangle whose side lengths are in the golden ratio Φ is called a goldenModelling rectangle optical. The golden lenses rectangles with could Dynamic be found in the Geometry compositions Software of artists and architectsAndreas Ulovec throughout history. Here are some examples:

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

2 Easy beginnings – light hits a plane surface 2.1 Reflection WeWhen usually a ray offind light the hits golden a plane ratio glass depicted surface, aas part a singleof it is largereflected. rectangle The law formed of reflection by a squaresays that and anotherthe angle rectangle. of incidence What (between is unique the ray about of light this and is the that normal we )can is equal repeat to thethe angle sequence of reflection: infinitely and perfectly within each section (Fig. 4).

Fig. 4 Golden rectangles

If we take away the big square on the left, what remains is yet another golden rectangle…and so on. Fig.1 Reflection of light at a plane surface. This project has been funded with support from the European Commission in its Lifelong Learning Programme (510028-LLP-1-2010-1-IT-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Task 5. Construct a sequence of golden rectangles in GeoGebra (as shown in Fig. 4). Modelling optical lenses with Dynamic Geometry Software Hint.Andreas Fist Ulovec construct a single golden rectangle as follows (Fig.5):

1.1 ConstructIntroduction a unit square (blue). 2. Draw a segment from the midpoint of one side to an opposite corner. 3.In Use optics that when segment it comes as thedown radius to show of an the arc path that of definesrays of lightthe long througher dimension glass, lenses of orthe systems rectangle. of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic valueFig. is 5 not Construction quite the same), of a butgolden complementing rectangle) it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an οoptical lens, a part of it gets reflected back in a certain angle, and Taskanother 6. partConstruct penetrates an arcthe ofglass 90 andin eachcontinues square there, so inas anotherto get the angle. golden The spiralsame happens in Fig. 6.when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

2 Easy beginnings – light hits a plane surface

2.1 Reflection When a ray of light hits a plane glass surface,Fig. 6 The a part golden of it is spiral reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection: Task 7. Construct the eyes of the golden rectangle (Fig. 7) and create a corresponding button for locating the eyes of a composition.

Hint: 1. Draw the diagonals of the rectangle. 2. From the center to each corner, construct the midpoint of every half-diagonal.

Fig.1 Reflection of light at a plane surface.

This project has been funded with suppoFig.rt from 7 Thethe European eyes of Commission a rectangle in its Lifelong Learning Programme (510028-LLP-1-2010-1-IT-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

These points—represented by the blue dots in the diagram above—are called the eyes of the rectangleModelling. optical lenses with Dynamic Geometry Software Andreas Ulovec The placement of the golden ratio intersections varies according to the proportions of the canvas.1 Introduction In the first format below, the golden sections divide the square canvas almost in thirds, whereas in the second one the lines fall closer to the centre: In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of Tasklight through8. Construct a lens with buttons the help based of dynamic on the geometry golden ratiosoftware and (DGS). explore the paintings in the folder Golden. Describe what you observe. This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

When describing a painting you might need a more specialized vocabulary. Here is something to start with: Art vocabulary Balance – can be symmetrical, asymmetrical, or radial Emphasis – the part of the painting that draws your attention Harmony – refers to the interrelationships of elements Movement – determined by the way the eye moves through the painting Proportion – the relationshipsFig.1 between Reflection elements, of light at including a plane surface. ratios such as the Golden Ratio Rhythm – the placement of elements to create a visual tempo or beat This project has been funded with support from the European Commission in its Lifelong Learning Programme (510028-LLP-1-2010-1-IT-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

VII. Dynamic mini-projects Modelling optical lenses with Dynamic Geometry Software 1.Andreas Take Ulovec a picture of a scenery in two ways so that they reflect specific goals. Explore the result by means of dynamic constructions and edit the pictures correspondingly by cutting out. 2.1 ArrangeIntroduction for a picture in two ways (according to two methods for composition): In optics. 6 personswhen it comes at a birthday down to p showarty sittingthe path around of rays a of round light tablethrough glass, lenses or systems of lenses,. amany class physics of 24 pupilsteachers and groan their – teacherthe experiments are quite complex, and you need a lot of equipment.. flowers It is difficultand fruits enough to show a ray of light in air – you need smoke, dust or any other way of making. perfumes light visible. and an To advertisement show the path of light in materials, you need special equipment – smoke Exploreglass lenses the etc.result Now with that’s dynamic not always constructions available, and and adjustments make corrections to the system if necessary. can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens 3.thicker, Make you an advertisementhave to take out in the two current ways lens of: and put in the new one. Students can then observe the situation. your before sch theool change and after the change – but it is not exactly a gradual change that lets them observe. your how thehobby path of light actually changes. We want to demonstrate how you can show the path of light. throughnatural a lens juices with the help of dynamic geometry software (DGS). This. materialan old can town be useful for science teachers, who can use it to model experiments with lenses, Explorereflection the and result refraction with dynamic– not instead constructions of the actual and experiment make corrections (if one seesif necessary. experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful 4.for Make mathematics in two teachers.ways a design Well, now of an where invitation is the mathematics? cart for: There is a lot of it in there! If a ray of light. hitsa festthe glassof mathematics surface of an (physics, optical lens, music, a part the of flowers it gets ,reflected athletics back) in a certain angle, and another. apart ball penetrates with masques the glass and continues there, in another angle. The same happens when the light. reachesa birthday the other party surface of the lens – again mathematics is required to calculate the angle in Explorewhich the the light result is ref withlected dynamic and refracted. constructions For ideal andlenses, make there corrections is an easy equationif necessary. calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in 5.near Create the centrea dynamic of the construction lens. With thicker in the lensesstyle of and the light artist being Max m Billore off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

6. Explore the rotational dynamic constructions by means of the sliders so as to create models similar to the pictures of rotational objects.

7. Create models of objectsFig.1 around Reflection you based of light on at rotational a plane surface. symmetry (wood carved ceilings, embroideredThis project has table been fundclothes,ed with etc.) suppo rt from the European Commission in its Lifelong Learning Programme (510028-LLP-1-2010-1-IT-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

8. Find paintings by the Bulgarian artist Ivan Milev and explore them by means of dynamic tools.Modelling optical lenses with Dynamic Geometry Software Hint:Andreas http://www.ivan-milev.com/ Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

Fig.1 Reflection of light at a plane surface.

Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

Fig.1 Reflection of light at a plane surface.

Modelling optical lenses with Dynamic Geometry Software Andreas Ulovec

1 Introduction In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan – the experiments are quite complex, and you need a lot of equipment. It is difficult enough to show a ray of light in air – you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment – smoke glass lenses etc. Now that’s not always available, and adjustments to the system can usually only be done by removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can then observe the situation before the change and after the change – but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of dynamic geometry software (DGS). This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction – not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflected back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens – again mathematics is required to calculate the angle in which the light is reflected and refracted. For ideal lenses, there is an easy equation calculating these effects – but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being more off-centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.

2 Easy beginnings – light hits a plane surface 2.1 Reflection When a ray of light hits a plane glass surface, a part of it is reflected. The law of reflection says that the angle of incidence (between the ray of light and the normal) is equal to the angle of reflection: Materials used Chehlarova, T., E. Sendova. Stimulating different intelligences in a congruence context. In: Constructionist approaches to creative learning, thinking and education: Lessons for the 21st century. Proceedings for Constructionism 2010. The 12th EuroLogo conference. 16-20 August, Paris, France. 2010. ISBN 978-80-89186-65-5 (Proc) ISBN 978-80-89186-66-2 (CD) http://dmentrard.free.fr/GEOGEBRA/art/ARTGEOGEBRA.htm http://jmora7.com/Arte/arte.htm http://www.dossantosdossantos.com/Arte/Arte_com_c%C3%B3nicas.html Recommended reading Stephen Skinner, Sacred Geometry, Sterling, New York/London, 2009 Priya Hemneway, Divine Proportion, Sterling Publishing, New York, 2005 Matila Ghyka, The geometry of art and life. New Dover publications, Ins, York Mario Livio, The Golden Ratio,Fig.1 ReflectionBroadway of Books, light at Newa plane York, surface. 2002 ScottThis project Olsedn, has beenThe fundGoldened with Secti suppoonrt, fromWalker&Company, the European Commission New York,in its Lifelong 2006 Learning Programme (510028-LLP-1-2010-1-IT-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.