#### second floor

### Pere Puig Adam Room

Geometry, curves, polyheders and inductive formulas

This room located in front of the staircase and elevator is dedicated to both flat and 3D geometry, curves and inductive formulas.

## Who was Pere Puig Adam?

Pere Puig i Adam (Barcelona, 1900 – Madrid, 1960) was a pedagogue, mathematician and engineer.

In addition to his research work mathematically modelling problems arising from physics, as a mathematics professor he wrote important textbooks in both the university and secondary spheres. His ideas ahead of his time are still a benchmark today. See for example his 1955 decalogue

*Not to adopt a rigid didactic, but adapted in each case to the student, observing it constantly.**Do not forget the specific origin of Mathematics or the historical processes of its evolution.**Present Mathematics as a unit in relation to natural and social life.**Carefully graduate abstraction plans.**Teach by guiding the student's creative and discovering activity.**Stimulate this activity awakening direct and functional interest towards the object of knowledge.**Promote self-correction as much as possible.**Achieve a certain mastery in solutions before automating them.**Take care that the expression of the student is a faithful translation of his thought.**To provide any student with successes that prevent their demoralization.*

## Some modules in this room

The curves that are obtained by rolling circumferences of different sizes.

Look at the sections of objects illuminated by the red LEDs of this ring.

Three rings to link them in a very special way.

Try putting the tetraeder and octahedre in the cube.

Displaying numeric properties with buckets and other pieces.

Two balls, one low in a straight line, the other curving. What's the point before?

The artisanal wooden cone that shows its sections: Circumference, ellipse, parabola and hyperbola.

With the same pieces, it reconstructs two polygons.

Put the skyscrapers taking into account how many you see from each position.

The old known multiplication table, turned into sculpture.

A tiled tiles with tiles in the form of a lizard designed by the artist M. Escher.

Build polyhedras with magnetized pieces.

We raise these arches with pillows.

How do the lengths, surfaces and volumes of similar objects change?

With all the pieces we build 3, 2 or 1 equilateral triangles

Put the blue poles, perpendicular to the edges of the dodecaedre to build the icosaedre.

Multiple ways to visualize and understand this famous theorem

## PPA01 – The dual of the regular dodecaedron

By placing, thanks to magnets, the woods perpendicular to the edges of the dodecahedron form its dual polyhedron, the icosaedron.

Recommended age: +8 years

Keywords: PERPENDICULAR, EDGE, ICOSAEDR, POLYHEDR, GEOMETRY

## PPA02 – Ring of fire

The lasers located around the ring allow us to visualize with a red luminous line the different sections of the figure that we place inside. There are different geometric figures made of transparent material (cube, cone, cylinder, dodecaedre). It is the result of an exchange with Momath of New York.

Recommended age: –

Keywords: –

## PPA03 – Cycloid

Wooden apparatus with two tracks next to the other, one straight and the other in the form of inverted cycloid.

When releasing two balls at the same time it is verified that the route of the cycloid is faster than that of the straight line and that the time for a ball to arrive from any point of the cycloid at the center point is always the same.

Recommended age: +4 years

Keywords: CURVE, TAUTOCHRONE, BRACHISTOCHRONE, SKATING, ANALYSIS, GEOMETRY

## PPA04 – Perimeter, area and volume

This module is a sequential itinerary consisting of three tables.

The first proposes to compare the perimeter of a large circle with the perimeters of two smaller circles, the sum of whose diameters is that of the initial circle.

The second table asks to choose if it will pass more fluid by a tube of 20 cm in diameter than by two of 10 cm in diameter. The sections of the tubes allow comparing the areas and measuring them with chickpea units.

The last table consists of two scales. In the first it is about matching the weight of a large cylinder with cylinders in diameter half and the same height. In the second, equal a sphere with spheres half in diameter.

Recommended age: +10 years

Keywords: CIRCLE, CYLINDER, SPHERE, CIRCUMFERENCE, DIAMETER, HEIGHT, SCALE, WEIGHT, SIMILARITY, REASON, PROPORTION, SCALE, GEOMETRY

## PPA05 – Polygon dissections

A set of 6 replacement puzzles. With the pieces of each color it is necessary to form the polygons indicated with their silhouette.

Recommended age: +6 years

Keywords: PUZZLE, LENGTH, ANGLES, AREA, GEOMETRY

## PPA07 – Virtual image with 2 parabolic mirrors

This device creates a virtual image of the object located inside, consisting of two parabolic mirrors that exchange the spotlights.

Recommended age: +4 years

Keywords: OPTICS, HOLOGRAM, GEOMETRY

## PPA07 – Cone of Apollonius

Built by the artisan workshop of Francisco Treceño http://www.artmadera.com/. The 5 pieces of wood of different shades show the 4 sections of the cone: Circumference, ellipse, parabola and hyperbola.

Recommended age: +14 years

Keywords: CIRCUMFERENCE, EL· LIPSE, HYPERBOLA, PARABOLA, SECTION, SURFACE, CONIC, GENERATING, AXIS, GEOMETRY

## PPA08 – From octahedre to cube

These 21 pieces that can be easily joined with magnets, allow, from the octahedre, the successive construction of the tetraeder, the double tetraeder (or "Stella octangula") and the cube.

The last 12 pieces that make up the cube are painted in 4 different colors, 3 of each. They allow to pose two combinatorial problems, the first that the vertices of the cube have the same color and the second that on each side there are the four colors.

Recommended age: +8 years

Keywords: COMBINATORICS, POLYHEDRE, VOLUME, KEPLER, STELLA OCTANGULA, OCTAHEDR, TETRAHEDRE, HISTORY, GEOMETRY

## PPA10 – Hypocycloides and epicicloids

A board with a fixed circular gear and a set of toothed circular parts. When rolled inside you can see how the points on the edge follow the drawn hypocycloids. By rotating them on the outside, epicicloids are obtained.

Recommended age: +14 years

Keywords: ARISTOTLE, PTOLOMEUS, SPIROGRAPH, SPIROGRAPH, DELTOID, NEPHROID, ASTROID, CARDIOID, HISTORY, GEOMETRY

## PPA11 – Gear without teeth

The first part consists of a gear of 50 teeth inside which can rotate a circle of 25 teeth, the 5 color points of the circle follow the 5 diameters. The second part is a pentagonal piece with the 5 color vertices located in the same position as before. The pentagon can rotate gently thanks to the laces inserted in the trenches built as the diameters of the first part.

Recommended age: +10 years

Keywords: DIAMETER, MECHANISM, GEOMETRY

## PPA13 – Odd and Square

On this board you can place the L-shaped pieces representing odd numbers 1, 3, 5, 7, 9, 11, 13 and 15. It is displayed as each piece completes a square and therefore the property that the sum of consecutive odds starting with 1 is a square number.

Recommended age: +16 years

KeywordS: PUZZLES, SPEECHLESS DEMONSTRATION, VISUALIZATION, INDUCTION, FIGURED NUMBERS, ARITHMETIC AND NUMBER THEORY

## PPA14 – Suma cubes

Pieces of 3 colors with which the cubes of 3, 4 and 5 can be formed on the one hand and on the other hand the cube of 6

Recommended age: +16 years

Keywords: PUZZLE, SPEECHLESS DEMONSTRATION,

VISUALIZATION, INDUCTION, FIGURATIVE NUMBERS, ARITHMETIC AND

NUMBER THEORY

## PPA14 – Odd and cubes

Board with 15 three-dimensional pieces containing an odd number of units, from 1 to 29. It allows to check how with the corresponding consecutive pieces the cubes of 1, 2, 3, 4 and 5 are formed.

Recommended age: +16 years

Keywords: PUZZLE, SPEECHLESS DEMONSTRATION,

VISUALIZATION, INDUCTION, FIGURATIVE NUMBERS, ARITHMETIC AND

NUMBER THEORY

## PPA14 – Squares and cubes

Board with 17 pieces with which the cubes of 1,2,3,4 and 5 can be formed on one side of each color and on the other hand the square of 1+2+3+4+5

Recommended age: +16 years

KeywordS: PUZZLE, SPEECHLESS DEMONSTRATION, VISUALIZATION, INDUCTION, FIGURED NUMBERS, ARITHMETIC AND

NUMBER THEORY

## PPA14 – Sum of triangular numbers

It consists of six equal pyramids representing the sum of the triangular numbers 1+3+6+10+15. It is about forming an orthoedrine of dimensions 5,6 and 7

Recommended age: +16 years

KeywordS: PUZZLE, SPEECHLESS DEMONSTRATION, VISUALIZATION, INDUCTION, FIGURED NUMBERS, ARITHMETIC AND

NUMBER THEORY

## PPA14 – Sum of square numbers

It consists of 6 staggered pyramids the same. The steps have 1, 4, 9, 16 and 25 (the first square numbers) units. The challenge is to form an ortho warp of dimensions, 5, 6 and 11

Recommended age: +16 years

KeywordS: PUZZLE, SPEECHLESS DEMONSTRATION, VISUALIZATION, INDUCTION, FIGURED NUMBERS, ARITHMETIC AND

NUMBER THEORY

## PPA15 – Blurred Knot

It consists of three rings each consisting of two pieces that are joined with magnets. The challenge is to join the three rings so that they are tied and that only by unplugging any of them are free.

Recommended age: +10 years

Keywords: KNOT THEORY, BORROMINI, LINK, TOPOLOGY

## PPA15 – Three equilateral triangles

It consists of 5 pieces of wood with which three challenges are proposed: Build three equilateral triangles, build two of them and - the third challenge - build only one.

Recommended age: +10 years

Keywords: PROPORTION, SCALE, AREA, GEOMETRY PUZZLE

## PPA16 – Pythagorean dissections

Set of panels with magnetized pieces that allow to check the Pythagoras theorem. On each panel, with the same pieces it is necessary, on the one hand, to form the square of the hypotenuse and on the other, to form the two squares on the catets.

Recommended age: +10 years

Keywords: AREA, STAIRCASE, PYTHAGORAS, HYPOTENUSE, CATET, TRIANGLE, HISTORY, GEOMETRY

## PPA17 – Pythagorean disappearance

The four pieces can be placed in the frame of two shapes, one, fill the frame and on the other they leave a small central square.

Recommended age: +14 years

Keywords: PYTHAGORAS, HYPOTENUSE, CATET, TRIANGLE, GEOMETRY

## PPA18 – Pythagorean balance

The balance allows us to verify that the weight of the square corresponding to the hypotenuse of the rectangle triangle is the same as the sum of the weights of the two squares corresponding to the catets.

Since here the weight is proportional to the area, it is an empirical check of the Pythagoras theorem.

In addition, it also allows us to verify that the same relationship is valid for the set of 3 triangles or 3 similar semicircles, since their linear dimensions also correspond to the sides of the rectangle triangle.

Recommended age: +14 years

Keywords: AREA, STAIRCASE, PYTHAGORAS, HYPOTENUSE,

CATHET, TRIANGLE, GEOMETRY

## PPA19 – Pantograph

Pantograph model that allows you to follow similar figures of 1:2 proportion.

Recommended age: +14 years

Keywords: TRANSFORMATION, PROPORTION, HOMOTECHNICS, SCALE, SIMILARITY, GEOMETRY

## PPA20 – Investor

Articulated apparatus showing the transformation of the plane called geometric inversion. Following with one end the silhouettes of straight lines and circumferences is checked how the other end draws the corresponding figures.

Recommended age: +16 years

Keywords: TRANSFORMATION, SCALE, PRODUCT, SEGMENT

16, GEOMETRY

## PPA20 – Two-triangle symmetries

Two equal triangles, of different colors on each side, not symmetrical. One of them is divided into three parts which are symmetrical. 4 challenges are proposed consisting of their placement in the corresponding spaces.

Recommended age: +14 years

Keywords: SYMMETRY, BISECTRIC, INCENTIVE, GEOMETRY

## PPA20 – Rotors, triangle de Reuleaux

There are 3 objects in this module. The first is a triangle of Reuleaux, which can be seen rotating within a square touching all four edges. The second is the spindle-shaped figure that, equivalently, can rotate inside an equilateral triangle. The third object is the model of a revolving door that instead of rotating in a cylindrical space, as usual, rotates within a square prism.

Recommended age: +14 years

Keywords: CURVE, PI NUMBER, CONSTANT WIDTH, GEOMETRY

## PPA21 – Escher's lizards

Six pieces in the form of a lizard, reproduction made in 3D printing of those that appear in the work Reptiles de M.C. Escher. They allow you to check how they tiles the plane.

Recommended age: +4 years

Keywords: TILING, ART, GEOMETRY

## PPA22 – Fitting polyheders

It consists of a tetraeder and a wooden octaheder that must be fitted into a cube.

Recommended age: +8 years

Keywords: SPACE, DIAGONAL, SECTION, GEOMETRY

## PPA24 – Polydron

Material consisting of triangles, squares, pentagons and hexagons that can be joined with magnets along the edges. It allows multiple constructions and geometry workshops.

Recommended age: +6

Keywords: 3D STRUCTURES

## PPA25 – Semicircular arch

To build the semicircular arch there are 9 padded pieces equal. On the table two woods fastened with Velcro prevent it from falling.

Recommended age: +10

Keywords: CIRCLE, ARCHITECTURE,

ROMANESQUE, BUTTRESS, TECHNOLOGY, GEOMETRY

## PPA26 – Catenary Arc

To build this catenary arch there are 11 colored padded pieces and a ribbon or wooden mold.

Recommended age: +10

Keywords: CURVE, ARCHITECTURE, GAUDÍ, TECHNOLOGY, GEOMETRY

## MA06 – The Heart of Archimedes

By turning the wheel, two ways can be observed to transform the circular movement into a rectilinear movement of vaivé and how the heart of Archimedes serves to make it of constant speed.

Recommended age: +14 years

Keywords: CURVE, CIRCULAR MOVEMENT, LINEAR MOVEMENT, SYMMETRY, HISTORY, GEOMETRY

## MA06 – Lamp table

By superimposing translucent pieces of different sizes and colors, we must obtain the requested figures.

Recommended age: +8 years

Keywords: GEOMETRIC FIGURES

## Additional resources and material

- Dossier to work in this room prepared by Toni Gomà and Daniel Bosch
- Didactic sheets of Marina Agudelo, 7 correspond to modules in this room
- Didactic sheets of ADEMGI colleagues, 4 correspond to modules in this room.
- Article by Josep Rey and Manel Udina in NouBiax 38 on the inductive formula module.
- Article by Anton Aubanell at NouBiax 39 on the module "Perimeter, surface,volume"