Towards meaningful learning

 

Leonardome is an innovative educational activity for you and your students. We offer you the opportunity to construct domes based on the various patterns designed by Leonardo da Vinci 500 years ago. It is a multi-disciplinary and playful activity in which knowledge, experimentation and teamwork all come together.

The domes of Leonardome are built using pieces which are all the same shape and without any fixing element: the weight of the pieces themselves is sufficient to maintain its stability.

The construction makes it possible to work with multiple concepts and abilities. This is why we talk about meaningful learning: knowledge is assimilated in an integrated way and, at the same time, the competencies that are the basis of 21st century education are developed.

With this resource, we put the students at the centre of their own learning, we allow them to experiment and, no less importantly, we awaken their emotions.

Chapter 1

What will we learn? Key concepts and knowledge

Although the basis of activity is geometry, this resource is suitable for developing educational projects in fields such as history, art, architecture and technology. Leonardome offers a clear example of cross-curricular learning as it sets out horizontal connections between multiple areas of knowledge.

  • Mathematics
    We develop the capacity for reasoning and the power of abstract thinking. We describe spatial relationships and apply geometric transformations. We use visualisation and geometric models to solve problems. Utilizamos la visualización y los modelos geométricos para resolver problemas.
  • Technology
    The dome is a technical solution to a roofing system. It is a stable and supportive structure. It involves the basic laws of the statics and the resistance of the materials.
  • History
    It is based on a historical document (the 'Codex Atlanticus') by Leonardo da Vinci. It teaches you to appreciate the various artistic productions and put them into the context of each historical moment. We use visualisation and geometric models to solve problems.
  • Art
    We will discover Leonardo as a Renaissance artist. The dome has formed part of buildings throughout the history of art. Geometric patterns have inspired contemporary artists such as Rinus Roelofs and Buckminster Fuller.
Chapter 1.2

Leonardome allows to work the following skills

In particular, Leonardome allows the development of the following skills:

MATHEMATICAL
ARTISTIC AND CULTURAL
SCIENTIFIC- TECHNOLOGICAL
AUTONOMY AND INITIATIVE
COMMUNICATIVE
INTERACTIVE WORLD
Chapter 2

Leonardo da Vinci: Archetypal man of the Renaissance

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Genius, polymath, ahead of his time, homo universalis… These and many other terms define the figure of Leonardo da Vinci, the archetype of the Renaissance man. More than six thousand pages of his notebooks have been preserved, which bring together research into various disciplines such as anatomy, mechanics, engineering and optics.

Apart from his pictorial work (great but not plentiful), Leonardo was passionate about observation, experimentation and the exact formulation of general principles based on empirical experience. In his notebooks, those who have studied them have discovered, surprisingly, ideas and proposals that anticipate later developments in modern science, especially in the field of machines and engineering works.

Leonardo was the archetypal man of his time, a true humanist of the Renaissance.

Capítulo 2.1

Renaissance humanism

I do not think of becoming a god, nor of having the gift of eternity […] I am mortal and I only wish for mortal things (Petrarch)

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_Detail of the School of Athens, a fresco painting by the Renaissance artist Rafael (Apostolic Palace of the Vatican)

A new way of thinking for a new society. Between the 14th and 16th centuries, the concept of “Renaissance” began the Modern Age, which involved a break with the medieval worldview and reflected the ideals of the humanist movement. The phrase “man is the measure of all things” expresses the attitude of the new ruling class – the bourgeoisie – very well in its approach to life. Its philosophy was anthropocentric

(man is the centre of the universe and the highest achievement of nature) and was more interested in the world than in spiritual matters. The philosophers looked to antiquity and defended religious tolerance. As explained by Professor Vicente Moreno, “Humanism was an intellectual movement destined to transform medieval intellectual structures to adapt them to a more open and dynamic type of society”.

In this context of profound change, science advanced and figures of the first order emerged, such as the astronomers Copernicus and Galileo and Leonardo himself. Class (almost) wasn’t important: man had transformed himself into a being capable of observing, analysing, changing and mastering his environment.

Chapter 2.2

Leonardo through his artistic work

Beauty perishes in life, but it is immortal in art (Leonardo da Vinci)

Born in the Tuscan town of Vinci, Leonardo was a painter, anatomist, architect, scientist, writer, sculptor, philosopher, engineer, inventor … and, even, a musician, poet, town planner and palaeontologist. A lover and observer of nature, his great curiosity was a driving force for his life and work.

Leonardo received his first training in the workshop of the painter and sculptor Verrocchio. From this time one of his first well-known works is conserved, Arno Valley Landscape or the Landscape of Santa Maria della neve (1473), a drawing made with pen and ink. His career as a painter began with works of great quality such as The Annunciation (1472-1475) and improving the technique of the sfumato (fusion of light and shadow in very diffused colour tones) at a point of refinement never achieved until then.

Once in Milan and under the protection of the powerful Sforza family, Leonardo worked on architectural projects such as the dome of the cathedral and the modifications to the cathedral at Pavia. At the pictorial level, the most important production of his Milanese period is the Portrait of Cecilia Gallerani or Lady with an Ermine (Czartoryski collection in the National Museum of Krakow), The Virgin of the Rocks (1483-93; The Louvre, Paris) and The Last Supper for the refectory of the convent of Santa Maria delle Grazie (1499).

_In order: La Gioconda (The Mona Lisa), The Virgin of the Rocks, sketch of Leda and the Swan and Portrait of Cecilia Gallerani

After the French occupation of Milan, Leonardo began a journey that would take him to Mantua, Venice and Florence. The most representative artistic works of this period are the Portrait of Isabella d’Este (The Louvre), the cartoon of The Virgin and Child with St Anne and St John the Baptist, (1501; Royal Academy of Art, London), La Gioconda or Mona Lisa (started in 1503; Musée Du Louvre) and Leda and the Swan (Gallotti Spiridon Collection, Rome). The latter became the main model for the figura serpentinata (serpentine figure) composition, a rule that became the paradigm in painting during the 16th century.

After a brief return to Milan and a period in Rome (where he began his Trattato della pittura or Treatise on Painting) Leonardo departed for Le Clos-Lucé, summoned by his new patron and protector, Francis I, the king of France. The Florentine genius became the “first painter, first engineer and first architect of the king” and his activity was focused on the scientific field.

He was at the French court when he died in May 1519, the man who, along with Michelangelo and Raphael, has been considered one of the greatest Renaissance artists; and the universal man who observed and drew the world in order to understand it and make it beautiful and useful. To achieve, ultimately, the identity between art and science; the unity of knowledge.

Chapter 2.3

Leonardo, scientist

Our greatest follies may be very wise (Leonardo)

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_Design for a helicopter. Bortolon, The Life and Times of Leonardo, Paul Hamlyn. DP

Leonardo da Vinci was not just a great painter and draughtsman. In fact, in terms of quantity, his artistic output is less than his scientific work. Leonardo’s studies in science and engineering are astonishing and were recorded in numerous notebooks.

Self-taught and an untiring observer of natural phenomena, Leonardo drew everything that aroused his curiosity. From the notebooks we discover a wide variety of interests: composition of paintings, studies of details, animals, dissections, facial expressions, war machines, flying machines … Always conceiving science and art as a whole.

From fluid dynamics (of which Leonardo was a pioneer) to the anatomy of the heart, muscles and tendons; from mathematics to botany, from design to engineering… his intellectual voracity knew no limits.

In geometry, Leonardo built on the works of Euclid and Archimedes. Notable within this discipline is his analyses of the centres of gravity of geometric figures; the study of transformations of shapes into others while conserving the same volume; and the emerging empirical study of curved surfaces.

Chapter 2.4

The Codex Atlanticus by Leonardo

This compilation of documents contains the most spectacular drawings made by Leonardo on innovative technological projects. In its 1,119 pages, made up of sheets and fragments of all kinds and sizes, the detail and genius of Leonardo is reflected, as well as his keen interest in mechanics, anatomy and urban design.

The Codex Atlanticus brings together theoretical and practical aspects of painting and sculpture, on optics, perspective, the theory of light, shadows and materials used by the artist. In addition, it lists examples of architectural and urban projects that were never finished, such as the first studies for the renovation of the city of Milan or the project for a new Medici-Riccardi palace in Florence.

The reciprocal frames

Estudios sobre puentes y cúpulas
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_Self-supporting bridge that appears in Codex Atlanticus (f. 69ar and 71v). (www.leonardodigitale.com)

In the Codex, we also find the plans and drawings that brought about the Leonardome activity: Leonardo worked on the concept of building bridges and domes through the mutual support of the components. There are very few specifics on this idea, but one of these was the famous bridge built with batons without using any fixing material which can be seen on sheet 71v of the Codex Atlanticus.

It is here, therefore, that we find the principle of Leonardome: the mutual support of multiple components which allows structures to be created. This idea, born five hundred years ago, is known architecturally as a “reciprocal frame”: that is, a three-dimensional assembly of components that support each other. It has been used mainly in Eastern architecture.

 

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_Sheets 899v and 899r of the Codex Atlanticus with various geometric patterns (www.leonardodigitale.com)
Chapter 3

Rinus Roelofs: mathematics as an artistic inspiration

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A contemporary artist specialising in sculpture and mathematics, the Dutchman, Rinus Roelofs, could embody the continuation of the philosophy of Renaissance humanism. Born in 1954, Roelofs investigates and experiments with new designs and structures, for which he has earned a huge reputation in both scientific and artistic circles.

His research has led him to study and be inspired by the figure of the great Leonardo da Vinci. And, in rediscovering the drawings of the geometric patterns in the Codex Atlanticus, he has investigated symmetries, transformations and their possible derivations.

The Museu de Matemàtiques de Catalunya (Mathematics Museum of Catalonia, MMACA), for educational purposes, has developed a design of these patterns, inspired by the work of Leonardo and the research carried out by Roelofs.

Surprise, understanding, creation

“Mathematical structures can be found everywhere. We continually observe them in our daily lives. The use of these structures as visual decoration is so common that we do not even see this as being mathematical. But the study of the properties of these structures and – in particular – the relationship between the different structures can pose questions for us. And these questions may form the start of interesting artistic explorations.”

Roelofs works with wood, metal, acrylic and paper and creates fascinating works based on geometrical experimentation. Some even compare him with the artist, M.C. Escher, another Dutchman.

“The process starts with surprise and it is about understanding what is seen. Resolving these issues leads to new ideas, inspirations and designs”.  A bit like Leonardo da Vinci, right?

Capítulo 4

Shall we start building?

We offer 11 patterns to build the domes of Leonardo:

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Number 1

This is the easiest: it consists of squares of two sizes.

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Number 2

This is one drawn by Leonardo. It is formed of rectangles and squares.

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Number 3

This is formed by equal rectangles placed in herringbone form. We see it in tiled roofs, for example.

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Number 4

Formed by hexagons surrounded by triangles. The other original example by Leonardo.

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Number 5

Formed by large hexagons and small triangles; it grows rapidly.

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Number 6

Formed by hexagons surrounded by parallelepipeds.

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Number 7

Formed by hexagons surrounded by large rhombuses and triangles. The posts are aligned in straight lines.

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Number 8

Formed by large hexagons separated by a border of triangles and rhombuses.

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Number 9

Formed by trapeziums in the form of arrows that alternate directions.

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Number 10

Formed by irregular hexagons and small triangles.

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Number 11

Formed by irregular hexagons, rhombuses and triangles.

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Chapter 5

Do you want more? Extension activities

A series of suggestions in order to understand how the dome functions and to enter into the surprising world of its inspiration, Leonardo da Vinci.

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Self-sustaining circle

A simple, practical and totally illustrative proposal of the principle of the domes:

Begin by forming a circle with all the participants facing right, start walking around at the same time as making the circle smaller so that everyone gets closer together. Follow each other closely so you are next to each other; make the circle even smaller by getting closer to your fellow participants and putting your hands on the shoulders of the person in front of you.

Now, slowly, we can bend our legs until we are sitting on the knees of the person behind us, while the person in front of us is sitting on our knees.
If we take care and do it in a coordinated way, we won’t fall and will realise that we are copying the same principle as in Leonardo’s domes: the different pieces support each other!

Mirror writing

Did you know that Leonardo da Vinci wrote most of his personal notes using mirror writing? This simple method of encryption consists in writing on paper in the opposite direction to that used by most people. Thus, the result is a mirror image of normal writing (“back to front”): it appears normal when it is reflected in a mirror.

It is believed that Leonardo used it to “camouflage” some of his texts, which could have been considered inappropriate for the time; it could also be related to the fact that he was left-handed.

An example is found in one of his best-known drawings: the Vitruvian Man. Do you know how to decipher the annotations?

We are all equal: the proportions of the human body

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_Drawing of The Vitruvian Man, at the Accademia Galleries in Venice. PD

The famous Vitruvian Man is the theme of the following suggestion. The drawing represents a male figure with his arms and legs in two, super-imposed positions, and is inscribed within a circle and a square.

We are confronted with a study of the proportions of the human body, based on the treatise De Architectura (On Architecture) by Vitruvius, an architect of ancient Rome. It is also known as the Canon of Proportions of Man.

According to Leonardo himself, the height of a person is exactly the length of the extended arms (span). Do you want to check this?

Mark your height on a wall, stretch your arms and see how, while touching the ground, you can reach this point exactly.

Study of the proportions of the human body: did you know that…

Leonardo's parachute: a risky invention!

Leonardo was fascinated with the flight of birds: his writings are full of drawings of wings, flying machines and even parachutes.

We suggest you compare the time two sheets of paper of the same size take to fall. One of them will be flat and the other, rolled up in the form of a ball. What happened?

If the time it takes only depends on the weight, they should fall at exactly the same time, as would happen in an airless environment (for example, on the moon or inside a vacuum chamber).

Leonardo was one of the first to understand that the air has weight and volume.

And he drew a parachute in the form of a pyramid.

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_Model of the parachute designed by Leonardo. Nevit Dilmen – CC BY-SA
Are you up for building a model out of card?

 

You need: Sheet of A4 card of 120 g/m2, thread, paper clips, scissors, glue, ruler, compass.

Draw the following figure formed by 4 equilateral triangles and a flap:

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Instructions:

  • Cut along the continuous lines.
  • Fold along the dotted lines.
  • Make 4 holes in the places indicated by stars.
  • Glue the flap.
  • Put two pieces of thread through the holes and tie them.
  • Use a paper clip as a weight. If necessary, modify the length of the threads. The clip must be centred.

The moment of truth has arrived: let the parachute fall from a height. What happens?

Put on more clips; does it work better?

Undoubtedly, the parachute will overturn. The reason is that the air inside the pyramid has to get out somehow and, as it does so by the sides, it creates forces that make it capsize. This is why many parachutes have a hole in the centre to allow the air to flow without creating lateral forces.

Cut the vertex of the pyramid so that there is a hole of approximately 1 cm squared. Try the parachute now.

You can try different sizes for the central hole: the bigger it is, the less it turns. The problem is that the parachute falls faster because we have decreased the surface area.

You can experiment with different parachute designs, for example, making the pyramid flatter or making more holes.

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