Of the 32 polyores, 20 are convex and 12 are concave.

The goal of this exercise is to surround each convex polyore using 10 angle-edges and a few straight edges ; of these 20 convex polyores, 12 have sides that measure either 1 or φ, as below :

The other 8 can have a side that measures 2, φ + 1 ou 2φ, as below :

Try to circle the 20 convex polyores.

Exercise 2 :

In this exercise, we are going to surround the concave polyores ; when we surround the polyore below, there is an empty space (hatched in drawing a) :

This empty space can be filled by 2 unit triangles of area 1 (drawing b) or by a polyore of area 2 (drawing c).

2 is the degree of concavity of this polyore ; we will say that this polyore is "2-concave".

In the following example, we have surrounded the polyore (drawing a) but the empty space has been filled with 2 small black triangles that can be found on the tray of La Ora Stelo (drawing b) :

These triangles are not polyores so we have to find another fence :

Drawing c shows another fence, and this time the empty space can be filled with 2 unit triangles of area φ (drawing d) or by a polyore of area 2φ (drawing e).

This polyore is "2φ-concave".

Try to find the degree of concavity of the 12 concave polyores.

Note: a convex polyore is "0-concave".

Exercise 3 :

Let's re-take the second polyore surrounded in Exercise 1 but now let's count the angle-edges :

If we start at angle 3, clockwise we have 3412 and counterclockwise 3214.

If we start at angle 4, we have 4123 or 4321.

If we start at angle 1, we have 1234 or 1432.

If we start at angle 2, we have 2341 or 2143.

We get 8 numbers ; the smallest of these 8 numbers is 1234 : we will say that this polyore is a "1234 quadrilateral".

1234 is its "code of angles" (angle code).

Let's take another example with the first polyore surrounded in Exercise 1 :

Using the same process, we find that its angle code is 1333.

Try to find the angle code of the 12 convex quadrilateral polyores.

Exercise 4 :

If you did Exercise 3 you found 7 different angle codes.

Before moving on to Exercise 4, look at the following shapes :

These shapes come from Exercise 2 ; they are respectively a 12223 pentagon and a 12313 pentagon : these shapess are made up of 2 polyores.

Exercise 4 is to find two convex quadrilaterals (obtained using 2 polyores each) with angle codes 1243 and 2224.

Exercise 5 :

Try to find the angle codes of the 4 polyores which are triangles and the 4 polyores which are convex pentagons.

Exercise 6 :

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For this exercise, you must use kits 1, 2 and 3 !

The draw below shows a concave polyore with its fence using an angle of 216° :

The aim of this exercise is to circle all concave polyores ; with this above, there are 12 in all ; but one of them has an angle of 324° and cannot be circled.