PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS

Size: px
Start display at page:

Download "PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS"

Transcription

1 PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS JORGE REZENDE. Introduction Consider a polyhedron. For example, a platonic, an arquemidean, or a dual of an arquemidean polyhedron. Construct flat polygonal plates in the same number, shape and size as the faces of the referred polyhedron. Adjacent to each side of each plate draw a number like it is shown in figure. Some of the plates, or all, can have numbers on both faces. We call these plates, two-faced plates. In this case, they have the same number adjacent to the same side. In figure, c and d are the two faces of a pentagonal two-faced plate example. Now the game is to put the plates over the polyhedron faces in such a way that the two numbers near each polyhedron edge are equal. If there is at least one solution for this puzzle one says that we have a polyhedron puzzle with numbers. These puzzles are a tool in teaching and learning mathematics, and a source of examples and exercises. They may be used in various fields such as elementary group theory and computational geometry. It is obvious that not all puzzles are interesting. But, amazingly or not, there are natural ways of constructing interesting puzzles. From now on, assume that the numbers belong to the set {,,..., n}, and that all the numbers are used. If we have plate faces which have the shape of a regular polygon with j sides, one can ask how many possible ways ν are there to draw the numbers,,..., n, without repeating them on each plate face. The answer is ( ) n ν = (j )! = j n! (n j)!j. For n = and j = (equilateral triangle), this gives ν =. The Mathematical Physics Group is supported by the Portuguese Ministry for Science and Technology. 000 Mathematics Subject Classification. B0, B, 0B0, 0B0.

2 JORGE REZENDE For n = and j =, this gives ν = 8. Note that 8 is precisely the number of the octahedron faces. We naturally call the related puzzle, the octahedron puzzle. a b c d Figure. For n = and j =, this gives ν = 0. Note that 0 is precisely the number of the icosahedron faces. We call the related puzzle, the icosahedron first puzzle (or icosahedron ()). For n = and j =, this gives ν = 0. Note that 0 is the double of the number of the icosahedron faces. Construct different plates with the numbers written on both faces. This gives 0 plates. We call the related puzzle, the icosahedron second puzzle (or icosahedron ()). Consider again n = and j =. Construct different plates with the numbers written only on one face, but in such a way that the numbers grow if we read them, beginning with the minimum, counter clock-wise. See an example in figure b. This gives 0 plates. We call the related puzzle, the icosahedron third puzzle (or icosahedron ()). For n = and j = (square), this gives ν =. Note that is precisely the number of the cube faces. We naturally call the related puzzle, the cube puzzle. For n = and j = (regular pentagon), this gives ν =. Note that is precisely the double of the number of the dodecahedron faces. Construct different plates with the numbers written on both faces. This gives plates. In figure, c and d show the two faces of a plate of this type. We call the related puzzle, the dodecahedron first puzzle (or dodecahedron ()).

3 PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS Let again n = and j =. Construct different plates with the numbers written only on one face, but in such a way that the numbers read counter clock-wise, abcd, are such that abcd form an even permutation. This gives plates. In figure, c and d show two plates of this type. We call the related puzzle, the dodecahedron second puzzle (or dodecahedron ()). Consider n =. With j =, one has ν = 8. With j =, one has ν =. Note that 8 is precisely the number of the cuboctahedron triangular faces and is precisely the number of its square faces. This is an example of an interesting puzzle using an arquemidean polyhedron. Take now a deltoidal icositetrahedron. It has deltoidal faces. If we have plates which have the deltoidal shape the number of possible different ways to draw the numbers,,,, without repeating them on each plate is precisely. This an example of an interesting puzzle using a dual of an arquemidean polyhedron. These are simple examples of polyhedron puzzles with numbers, which are enough in order to understand the following sections. There are, obviously, others. For more examples see Reference [], which is a development of Reference []. Reference [] is a collection of some of these puzzles paper models.. Polyhedron symmetries and permutation groups Consider a puzzle with numbers,,..., n drawn on the plates. From now on P denotes the set of its plate faces which have numbers drawn, and call it the plate set. S n denotes the group of all permutations of {,,..., n}; σ S n means that σ is a one-to-one function σ : {,,..., n} {,,..., n}. The identity is σ 0 : σ 0 () =, σ 0 () =,..., σ 0 (n) = n. The alternating group, the S n subgroup of the even permutations, is denoted by A n. We shall use also the group {, } S n denoted by S ± n. If δ, δ {, } and σ, σ S n, then (δ, σ ) (δ, σ ) = (δ δ, σ σ ). We denote S + n = {} S n S n. If Λ is a set, then Λ denotes its cardinal. Hence, if G is group, G denotes its order. From now on E denotes the set of the polyhedron edges and F denotes the set of the polyhedron faces. A solution of the puzzle defines a function ε : E {,,..., n}. Denote E the set of these functions. One can say that E is the set of the puzzle solutions. Denote Ω the group of the polyhedron symmetries. Every symmetry ω Ω induces two bijections, that we shall also denote ω, whenever there is no confusion possible: ω : E E and ω : F F. Denote also

4 JORGE REZENDE Ω {ω : E E} {ω : F F }, the two sets of these functions. One can say that each one of these two sets Ω is the set of the polyhedron symmetries. With the composition of functions each one of these two sets Ω forms a group that is isomorphic to the group of the polyhedron symmetries. If ω, ω Ω, we shall denote ω ω ω ω. If Ω is a subgroup of Ω, then Ω acts naturally on the face set, F : for ω Ω and ϕ F, one defines the action ωϕ = ω (ϕ). In the following Ω + denotes the subgroup of Ω of the symmetries with determinant. We shall also consider the group S n Ω. If (σ, ω ), (σ, ω ) S n Ω, one defines the product (σ, ω ) (σ, ω ) = (σ σ, ω ω ). On group theory we follow essentially references [] and []... The plate group. Some S n subgroups act naturally on P. Let π P and σ S n. Assume that a, b, c,... are drawn on π, by this order. Then σπ is the plate face where the numbers σ (a), σ (b), σ (c),... are drawn replacing a, b, c,.... Let s S n ± and π P. If s = (, σ) σ, then sπ = σπ. If s = (, σ) σ, then sπ is a reflection of σπ. In this last case, if the numbers a, b, c,... are drawn on π, by this order, then sπ is a plate face where the numbers..., σ (c), σ (b), σ (a) are drawn by this order. The plate group, G P, is the greatest subgroup of S n ± that acts on P. If s S n ± and sπ P, for every π P, then s G P... The solution group. Let ε : E {,,..., n} be a solution of the puzzle. The group of this solution, G ε, is a subgroup of S n Ω; (σ, ω) G ε if and only if σ ε ω = ε. Denote Ω ε the following subgroup of Ω: ω Ω ε if and only if there exists σ S n such that (σ, ω) G ε. Note that if ω Ω ε there exists only one σ S n such that (σ, ω) G ε. From this one concludes that ω (σ, ω) defines an isomorphism between Ω ε and G ε. On the other hand (det ω, σ) G P. This defines g ε : Ω ε G P, g ε (ω) = (det ω, σ), which is an homomorphism of groups. If all the plates are different, as is the case in the given examples and as we shall assume from now on, (det ω, σ) defines completely ω. Hence, g ε establishes an isomorphism between Ω ε and g ε (Ω ε ) G P. Denote G P ε g ε (Ω ε ). Finally, G ε and G P ε are isomorphic. We can identify (σ, ω) with (det ω, σ), and G ε with the subgroup G P ε of G P. One can define a function u ε : F P, that associates to every face ϕ the plate face π that ε puts in ϕ. Define also P ε = u ε (F ). Note that

5 PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS P ε = F and that G P ε acts on P ε. Then one has for every ϕ F. (det ω, σ) (u ε ω (ϕ)) = u ε (ϕ),.. Equivalent solutions. Let ε, ε : E {,,..., n} be solutions of the puzzle. One says that these solutions are equivalent, ε ε, if there are ω Ω and σ S n, such that Note that (det ω, σ ) G P. Then, σ ε ω = ε. (σ, ω) ( σ σσ, ω ωω ) = (σ, ω ) (σ, ω) (σ, ω ) defines an isomorphism between G ε one has that s σ sσ and G ε. As det ( ω ωω ) = det ω, defines an isomorphism between G P ε and G P ε. If σ = σ 0 and det ω =, what distinguishes the solutions ε and ε is only a rotational symmetry. In this case ε ω = ε expresses another equivalence relation, ε ε. When we make a puzzle, in practice, we do not recognize the difference between ε and ε. We shall say that they represent the same natural solution, an equivalence class of the relation. Let ε, ε, ε E. As ε ε and ε ε, implies ε ε, one can say that the natural solution represented by ε is equivalent to ε. For ε E, represent by [ε] the set of natural solutions equivalent to ε. Choose now ω Ω, such that det ω =. For ε E and s = (δ, σ) G P, denote ε s = σ ε ω, where ω is the identity if δ = and ω = ω if δ =. The set {ε s : s G P } includes representatives of all natural solutions equivalent to ε. Then [ε] = G P G P ε. The cardinal of all the natural solutions is then given by [ε] G P G P ε, where the sum is extended to all different equivalence classes [ε].

6 JORGE REZENDE.. Equivalent actions. Consider two groups H and H that act on two sets K and K. For ρ j H j and x j K j, ρ j x j ( K j ) represents the action of ρ j on x j. One says that the action of H on K is equivalent to the action of H on K if there exists an isomorphism ξ : H H and a one-to-one function ζ : K K, such that for every ρ H and every x K one has ξ (ρ) ζ (x) = ζ (ρx). Note that if the action of H on K is equivalent to the action of H on K, and H is a subgroup of H, then the action of H on K is equivalent to the action of H = ξ (H ) on K. Assume that the action of H on K is equivalent to the action of H on K. Let K and K be the sets of the equivalence classes of such actions. Then there exists a bijection Z : K K, such that for every ρ H and every X K one has and ξ (ρ) Z (X) = Z (ρx) X = Z (X). These two conditions are necessary in order to have equivalent actions. Example.. If ε and ε are equivalent solutions of the polyhedron puzzle, the actions of G P ε and G P ε on P are equivalent. Let ξ (s) = σ sσ and ζ (π) = (det ω, σ ) π. Then ξ (s) ζ (π) = σ sσ (det ω, σ ) π = (det ω, σ ) sπ = ζ (sπ). Example.. If ε and ε are equivalent solutions of the polyhedron puzzle, the actions of Ω ε and Ω ε on F are equivalent. Let ξ (ω) = ω ωω and ζ (ϕ) = ω ϕ. Then ξ (ω) ζ (ϕ) = ω ωω ω ϕ = ω ωϕ = ζ (ωϕ). Example.. If ε is a solution of the polyhedron puzzle, the actions of Ω ε on F and of G P ε on P ε = u ε (P ) are equivalent. Let ξ (ω) = g ε (ω ) = (det ω, σ ) and ζ (ϕ) = u ε (ϕ). Then ξ (ω) ζ (ϕ) = ( det ω, σ ) u ε (ϕ) = (u ε ω (ϕ)) = ζ (ωϕ). Denote F = {F, F,..., F k } and P = {P, P,..., P k } be the sets of the equivalence classes of such actions. Then u ε induces a one-to-one function U : F P, U (F j ) = P j, F j = P j, and ( det ω, σ ) U (F j ) = U (ωf j ), for every j =,,..., k, ω Ω ε.

7 PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS 7 From now on we shall deal only with the action of a subgroup Ω of Ω on F, or the action of a subgroup G of G P on P. Hence, when we say that two groups H and H are equivalent (H H ), we are talking about these actions.. Examples Some of the examples we give in this section can easily be studied directly. It is the case of the cube, the octahedron and the dodecahedron () puzzles. We give also results for the dodecahedron (), icosahedron () and () puzzles. These results were obtained mostly with a computer. Similar results for the icosahedron () puzzle are left to the reader. There are good reasons for presenting results for these two icosahedron puzzles. The icosahedron () puzzle is mathematically rich, has a lot of natural solutions (over one million!). On the other hand, the icosahedron () puzzle is also very instructive because it has few natural solutions ( only ), which means that it is difficult to do it without the help of a computer... The cube puzzle. In this case there is only one equivalence class. G ε Ω + S. G ε =. G P = 8. If one puts a plate on a face then one has different possibilities for the other plates ( natural solutions): 8/. These two different possibilities and the equivalences G ε Ω + S can be used to translate the cube symmetries into permutations, in the same way as we shall do later with the icosahedron () puzzle and its canonical solution... The octahedron puzzle. There three equivalence classes. One can distinguish them in the following way. Take a solution. On every vertex of the octahedron, note the numbers that correspond to its four edges. There are two possibilities for a vertex: a) four distinct numbers; b) three distinct numbers, with one of them repeated. In the solution, count the number of vertices where the situation a) happens. They can be, or 0, that distinguish the three equivalence classes. The first class group is of order. The second class group is of order 8. The third class group is of order. As G P = 8, one has that if one puts a plate on a face then one has different possibilities for the other plates ( natural solutions): Although the cube and the octahedron are dual polyhedra, puzzles are not. Solutions can be dual. The first class of the octahedron puzzle is dual of the cube puzzle class.

8 8 JORGE REZENDE.. The dodecahedron () puzzle. This puzzle has three equivalence classes that we can distinguish in the following way. Consider a solution. On every dodecahedron vertex note the numbers that are on the edges around the vertex. There are 0 possibilities, but not all of them belong to the solution. There are some repetitions: 7 or. The solutions that have 7 repetitions on the vertices are equivalent. The solutions that have repetitions on the vertices belong to different equivalence classes. One of these classes has the repetitions on opposite vertices. The other has the repetitions on vertices that belong to the same edges. The first class group is of order 8. The second class group is of order. The third class group is of order. As G P = 0, one has that if one puts a plate on a face then one has ( 0 different possibilities for the other plates (0 natural solutions): ) The dodecahedron () puzzle. This puzzle has equivalence classes that can be distinguished in the following form. Consider two opposite dodecahedron edges. There are other four that are orthogonal to these two. The six edges are over the faces of a virtual cube where the dodecahedron is inscribed. There are five such cubes. The first equivalence class has the same number associated to the edges that belong to the faces of each cube. The second equivalence class has the same number associated to the edges that belong to the faces of one of the fives cubes. The first class group is of order 0 and the second class group is of order. As G P = 0, one has that if one puts a plate on a face then one has different possibilities for the other plates ( natural solutions): The icosahedron () puzzle. The icosahedron () puzzle has 9 equivalence classes: have groups of order, have groups of order, have groups of order, has a group of order, have groups of order, 0 have groups of order, has a group of order 8, have groups of order 0, have groups of order, have groups of order, has a group of order 0. As G P = 0, one has that once one puts a plate over a face there are 0 different possibilities (0 natural solutions): 0 ( There are two possible natural solutions with a group of order 0. One of them, that we call canonical solution, is dual of the dodecahedron () solution with a group of order 0. The group of these two solutions ).

9 PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS 9 z x y Figure. is equivalent to the icosahedron group: G ε Ω {, } A. The canonical solution is represented in figure... The icosahedron () puzzle. This puzzle has 97 equivalence classes: 90 have groups of order, and 7 have groups of order. As G P =, one has that, once one puts a plate over a face there are different possibilities ( natural solutions): Note that the actions of all these 7 groups of order are equivalent. Figure shows representatives of all these 7 equivalence classes.. More on icosahedron puzzles As we have already seen there is a natural solution of the icosahedron () puzzle which is dual of the dodecahedron () natural solution with a group of order 0 (see figure ). This group is equivalent to the icosahedron group Ω: G ε Ω {, } A. As this solution is very easy to construct, one can use it in order to translate in terms of

10 0 JORGE REZENDE Figure. {, } A everything that happens in Ω. For example, all subgroups of Ω are equivalent to the corresponding subgroups of {, } A... Subgroups of the icosahedron group. The icosahedron group Ω has different equivalence classes of subgroups: of order ({σ 0 }); of order ; of order ; of order ; of order ; of order ; of order 8; of order 0; of order ; of order 0; of order ; of order 0; of order 0 (Ω {, } A ). In the following a, b, c, d, e {,,,, } are different numbers, and in angle brackets we give the number of equivalence classes of solutions in the icosahedron () puzzle.... Groups of order. The equivalence classes are generated by (ab) (cd) 8, (, (ab) (cd)) and (, σ 0 ). Look at figure. The rotation around the z-axis by an angle of 80, corresponds to the permutation () (). This rotation belongs to the first equivalence class. The reflection using the xy-plane (orthogonal to the z-axis) as a mirror, corresponds to (, () ()). This reflection belongs to the second equivalence class. The third equivalence class corresponds to the central symmetry (x, y, z) (x, y, z).

11 PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS... Groups of order. The unique equivalence class is generated by (abc). In figure, the rotations around the axis x = y = z by angles multiple of 0 correspond to group generated by ().... Groups of order. Let σ = (ab) (cd) and σ = (ac) (bd). Note that σ σ = σ σ = (ad) (bc). The equivalence classes are generated by σ and σ 0, σ and (, σ 0 ), σ and (, σ ) 0, respectively.... Groups of order. The unique equivalence class is generated by (abcde). In figure, the rotations around the axis x = 0, z = + y (this is the z -axis in figure ), by angles multiple of 7, c orrespond to group generated by ().... Groups of order. Let σ = (ab) (cd) and σ = (ae) (cd). The equivalence classes are generated by σ and σ ( S ), (, σ ) and (, σ ) 0, (abc) and (, σ 0 ) 7, respectively.... Groups of order 8. Let σ = (ab) (cd) and σ = (ac) (bd). As before, note that σ σ = σ σ = (ad) (bc). The unique equivalence class is generated by σ and σ and (, σ 0 )...7. Groups of order 0. Let σ = (ab) (cd) and σ = (ac) (be). The equivalence classes are generated by σ and σ ( D, the dihedral group of order 0), (, σ ) and (, σ ) 0, (abcde) and (, σ 0 ), respectively...8. Groups of order. Let σ = (ab) (cd) and σ = (acd). The equivalence classes are generated by σ and σ ( A ), S and (, σ 0 ) ( {, } S ) 0, respectively...9. Groups of order 0. The unique equivalence class is generated by D and (, σ 0 ) ( {, } D ) Groups of order. The unique equivalence class is generated by A and (, σ 0 ) ( {, } A ).... Groups of order 0. The unique equivalence class is A Groups of order 0. The unique equivalence class is {, } A Ω.

12 JORGE REZENDE.. Subgroups of plate groups. All the subgroups of the icosahedron group have cyclic groups of order, or as generators. Let us look at the actions on the faces of such cyclic groups. The first equivalence class of the order groups has 0 orbits with elements and the determinant of the generator is : (, 0 ). The second equivalence class has 8 orbits with elements, orbits with element and the determinant of the generator is : (, 8 + ). The third equivalence class has 0 orbits with elements and the determinant of the generator is : (, 0 ). The equivalence class of the order groups has orbits with elements, orbits with element and the determinant of the generator is : (, + ). The equivalence class of the order groups has orbits with elements and the determinant of the generator is : (, ).... The icosahedron () puzzle. The plate group is G P = {, } S. The only cyclic groups of G P whose actions on the plates have orbits of the type (, 0 ), (, 8 + ), (, 0 ), (, + ) and (, ) are those generated, precisely, by (, (ab) (cd)), (, (ab) (cd)), (, σ 0 ), (, (abc)) and (, (abcde)).... The icosahedron () puzzle. Let σ = () and σ = () () (). The plate group, G P, is generated by (, σ ) and (, σ ). The only cyclic groups of G P whose actions on the plates have orbits of the type (, 0 ), (, 8 + ), (, 0 ) and (, + ) are generated by (, σ), ( ) ( ), σσ j,, σ k σ, (, σ ), j =,,, k = 0,,. There are no subgroups of order. It is easy to see directly that there are no solutions corresponding to the subgroups of order. The solutions corresponding to subgroups of order, are only those related to the generators ( ), σσ k, k = 0,,... Finding solutions of a puzzle with a prescribed group. A way to find a solution for a puzzle is to use the group relations between the polyhedron and the plate groups. Take, for example, the icosahedron () puzzle and the third equivalence class for groups of order 0. This class is generated by an element σ of order and (, σ 0 ). The numbers on the edges must respect the central symmetry and a rotational symmetry group of order. Let σ = (). In figure, that illustrates this example, a, b {,,,, }, and a j = σ j (a), b j = σ j (b), j =,,,. The numbers a and b must be chosen so that a b, a σ (b) = b, b σ (a) = a. These three conditions give 9 possibilities, but only of them correspond to solutions of the puzzle: (a, b) = (, ), (, ), (, ), (, ), (, ). The third one is the canonical

13 PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS z a a b b a b b b b a a Figure. solution. The first and the fourth ones belong to the same equivalence class. The same happens with the second and the fifth ones. These last two classes are the ones already listed. References [] Jorge Rezende: Puzzles numéricos poliédricos. Diário da República, III Série, n o 00, p. 7 (988). [] Jorge Rezende: Jogos com poliedros e permutações. Bol. Soc. Port. Mat., 0- (000). PT.html [] Jorge Rezende: Puzzles com poliedros e números. SPM: Lisboa 00. [] M. A. Armstrong: Groups and Symmetry. Berlin: Springer-Verlag 988. [] R. A. Wilson, R. A. Parker and J. N. Bray: ATLAS of Finite Group Representations. Grupo de Física-Matemática da Universidade de Lisboa, Av. Prof. Gama Pinto, 9-00 Lisboa, Portugal, e Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa address: URL: PT.html

1 Symmetries of regular polyhedra

1 Symmetries of regular polyhedra 1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

More information

The 5 Platonic solids:

The 5 Platonic solids: The 5 Platonic solids: The Tetrahedron (3 equilateral triangles at each vertex) The Hexahedron (3 squares at each vertex, cube) The Octahedron (4 equilateral triangles at each vertex) The Dodecahedron

More information

SOLIDS, NETS, AND CROSS SECTIONS

SOLIDS, NETS, AND CROSS SECTIONS SOLIDS, NETS, AND CROSS SECTIONS Polyhedra In this section, we will examine various three-dimensional figures, known as solids. We begin with a discussion of polyhedra. Polyhedron A polyhedron is a three-dimensional

More information

Star and convex regular polyhedra by Origami.

Star and convex regular polyhedra by Origami. Star and convex regular polyhedra by Origami. Build polyhedra by Origami.] Marcel Morales Alice Morales 2009 E D I T I O N M O R A L E S Polyhedron by Origami I) Table of convex regular Polyhedra... 4

More information

Chapter 7. Permutation Groups

Chapter 7. Permutation Groups Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

More information

(Q, ), (R, ), (C, ), where the star means without 0, (Q +, ), (R +, ), where the plus-sign means just positive numbers, and (U, ),

(Q, ), (R, ), (C, ), where the star means without 0, (Q +, ), (R +, ), where the plus-sign means just positive numbers, and (U, ), 2 Examples of Groups 21 Some infinite abelian groups It is easy to see that the following are infinite abelian groups: Z, +), Q, +), R, +), C, +), where R is the set of real numbers and C is the set of

More information

S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P =

S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P = Section 6. 1 Section 6. Groups of Permutations: : The Symmetric Group Purpose of Section: To introduce the idea of a permutation and show how the set of all permutations of a set of n elements, equipped

More information

Topological Treatment of Platonic, Archimedean, and Related Polyhedra

Topological Treatment of Platonic, Archimedean, and Related Polyhedra Forum Geometricorum Volume 15 (015) 43 51. FORUM GEOM ISSN 1534-1178 Topological Treatment of Platonic, Archimedean, and Related Polyhedra Tom M. Apostol and Mamikon A. Mnatsakanian Abstract. Platonic

More information

2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE?

2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE? MATH 206 - Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of

More information

Shape Dictionary YR to Y6

Shape Dictionary YR to Y6 Shape Dictionary YR to Y6 Guidance Notes The terms in this dictionary are taken from the booklet Mathematical Vocabulary produced by the National Numeracy Strategy. Children need to understand and use

More information

3D shapes. Level A. 1. Which of the following is a 3-D shape? A) Cylinder B) Octagon C) Kite. 2. What is another name for 3-D shapes?

3D shapes. Level A. 1. Which of the following is a 3-D shape? A) Cylinder B) Octagon C) Kite. 2. What is another name for 3-D shapes? Level A 1. Which of the following is a 3-D shape? A) Cylinder B) Octagon C) Kite 2. What is another name for 3-D shapes? A) Polygon B) Polyhedron C) Point 3. A 3-D shape has four sides and a triangular

More information

A. 32 cu ft B. 49 cu ft C. 57 cu ft D. 1,145 cu ft. F. 96 sq in. G. 136 sq in. H. 192 sq in. J. 272 sq in. 5 in

A. 32 cu ft B. 49 cu ft C. 57 cu ft D. 1,145 cu ft. F. 96 sq in. G. 136 sq in. H. 192 sq in. J. 272 sq in. 5 in 7.5 The student will a) describe volume and surface area of cylinders; b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c) describe how changing

More information

Every 3-dimensional shape has three measurements to describe it: height, length and width.

Every 3-dimensional shape has three measurements to describe it: height, length and width. Every 3-dimensional shape has three measurements to describe it: height, length and width. Height Width Length Faces A face is one of the flat sides of a three-dimensional shape. Faces A cuboid has 6 flat

More information

Geometry Progress Ladder

Geometry Progress Ladder Geometry Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes

More information

Chapter 18 Symmetry. Symmetry of Shapes in a Plane 18.1. then unfold

Chapter 18 Symmetry. Symmetry of Shapes in a Plane 18.1. then unfold Chapter 18 Symmetry Symmetry is of interest in many areas, for example, art, design in general, and even the study of molecules. This chapter begins with a look at two types of symmetry of two-dimensional

More information

Angles that are between parallel lines, but on opposite sides of a transversal.

Angles that are between parallel lines, but on opposite sides of a transversal. GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

More information

Line Segments, Rays, and Lines

Line Segments, Rays, and Lines HOME LINK Line Segments, Rays, and Lines Family Note Help your child match each name below with the correct drawing of a line, ray, or line segment. Then observe as your child uses a straightedge to draw

More information

Geometry A Solutions. Written by Ante Qu

Geometry A Solutions. Written by Ante Qu Geometry A Solutions Written by Ante Qu 1. [3] Three circles, with radii of 1, 1, and, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to

More information

MA.7.G.4.2 Predict the results of transformations and draw transformed figures with and without the coordinate plane.

MA.7.G.4.2 Predict the results of transformations and draw transformed figures with and without the coordinate plane. MA.7.G.4.2 Predict the results of transformations and draw transformed figures with and without the coordinate plane. Symmetry When you can fold a figure in half, with both sides congruent, the fold line

More information

Teaching Guidelines. Knowledge and Skills: Can specify defining characteristics of common polygons

Teaching Guidelines. Knowledge and Skills: Can specify defining characteristics of common polygons CIRCLE FOLDING Teaching Guidelines Subject: Mathematics Topics: Geometry (Circles, Polygons) Grades: 4-6 Concepts: Property Diameter Radius Chord Perimeter Area Knowledge and Skills: Can specify defining

More information

12 Surface Area and Volume

12 Surface Area and Volume 12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

More information

Hyperbolic Islamic Patterns A Beginning

Hyperbolic Islamic Patterns A Beginning Hyperbolic Islamic Patterns A Beginning Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-2496, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

More information

Nets of a cube puzzle

Nets of a cube puzzle Nets of a cube puzzle Here are all eleven different nets for a cube, together with four that cannot be folded to make a cube. Can you find the four impostors? Pyramids Here is a square-based pyramid. A

More information

Computing the Symmetry Groups of the Platonic Solids With the Help of Maple

Computing the Symmetry Groups of the Platonic Solids With the Help of Maple Computing the Symmetry Groups of the Platonic Solids With the Help of Maple Patrick J. Morandi Department of Mathematical Sciences New Mexico State University Las Cruces NM 88003 USA pmorandi@nmsu.edu

More information

1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

1. Determine all real numbers a, b, c, d that satisfy the following system of equations. altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

More information

Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions

Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.

More information

Level 1. AfL Questions for assessing Strand Five : Understanding Shape. NC Level Objectives AfL questions from RF

Level 1. AfL Questions for assessing Strand Five : Understanding Shape. NC Level Objectives AfL questions from RF AfL Questions for assessing Strand Five : Understanding Shape NC Level Objectives AfL questions from RF Level 1 I can use language such as circle or bigger to describe the shape and size of solids and

More information

SOLID SHAPES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier

SOLID SHAPES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier Mathematics Revision Guides Solid Shapes Page 1 of 19 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier SOLID SHAPES Version: 2.1 Date: 10-11-2015 Mathematics Revision Guides Solid

More information

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line. Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

More information

SYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING

SYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING Daniel DOBRE, Ionel SIMION SYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING Abstract: This article discusses the relation between the concept of symmetry and its applications

More information

56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.

56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points. 6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which

More information

Platonic Solids. Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren't polyhedra). Examples:

Platonic Solids. Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren't polyhedra). Examples: Solid Geometry Solid Geometry is the geometry of three-dimensional space, the kind of space we live in. Three Dimensions It is called three-dimensional or 3D because there are three dimensions: width,

More information

Activity Set 4. Trainer Guide

Activity Set 4. Trainer Guide Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF SOLID FIGURES

More information

Math 475, Problem Set #13: Answers. A. A baton is divided into five cylindrical bands of equal length, as shown (crudely) below.

Math 475, Problem Set #13: Answers. A. A baton is divided into five cylindrical bands of equal length, as shown (crudely) below. Math 475, Problem Set #13: Answers A. A baton is divided into five cylindrical bands of equal length, as shown (crudely) below. () ) ) ) ) ) In how many different ways can the five bands be colored if

More information

Chapter 8 Geometry We will discuss following concepts in this chapter.

Chapter 8 Geometry We will discuss following concepts in this chapter. Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

More information

Date: Period: Symmetry

Date: Period: Symmetry Name: Date: Period: Symmetry 1) Line Symmetry: A line of symmetry not only cuts a figure in, it creates a mirror image. In order to determine if a figure has line symmetry, a figure can be divided into

More information

Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS

Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS 1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps

More information

Maximizing Angle Counts for n Points in a Plane

Maximizing Angle Counts for n Points in a Plane Maximizing Angle Counts for n Points in a Plane By Brian Heisler A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF

More information

ABEL S THEOREM IN PROBLEMS AND SOLUTIONS

ABEL S THEOREM IN PROBLEMS AND SOLUTIONS TeAM YYePG Digitally signed by TeAM YYePG DN: cn=team YYePG, c=us, o=team YYePG, ou=team YYePG, email=yyepg@msn.com Reason: I attest to the accuracy and integrity of this document Date: 2005.01.23 16:28:19

More information

(a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units

(a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units 1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units

More information

Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item 2) (MAT 360) Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

More information

Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

More information

Binary Operations. Discussion. Power Set Operation. Binary Operation Properties. Whole Number Subsets. Let be a binary operation defined on the set A.

Binary Operations. Discussion. Power Set Operation. Binary Operation Properties. Whole Number Subsets. Let be a binary operation defined on the set A. Binary Operations Let S be any given set. A binary operation on S is a correspondence that associates with each ordered pair (a, b) of elements of S a uniquely determined element a b = c where c S Discussion

More information

CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session. Geometry, 17 March 2012 CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

More information

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,

More information

UNIT H1 Angles and Symmetry Activities

UNIT H1 Angles and Symmetry Activities UNIT H1 Angles and Symmetry Activities Activities H1.1 Lines of Symmetry H1.2 Rotational and Line Symmetry H1.3 Symmetry of Regular Polygons H1.4 Interior Angles in Polygons Notes and Solutions (1 page)

More information

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A. 1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

More information

alternate interior angles

alternate interior angles alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

More information

Geometry Module 4 Unit 2 Practice Exam

Geometry Module 4 Unit 2 Practice Exam Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning

More information

Teaching and Learning 3-D Geometry

Teaching and Learning 3-D Geometry This publication is designed to support and enthuse primary trainees in Initial Teacher Training. It will provide them with the mathematical subject and pedagogic knowledge required to teach 3-D geometry

More information

Geometric Transformations

Geometric Transformations Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

More information

CHAPTER 1. LINES AND PLANES IN SPACE

CHAPTER 1. LINES AND PLANES IN SPACE CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given

More information

Paper Folding and Polyhedron

Paper Folding and Polyhedron Paper Folding and Polyhedron Ashley Shimabuku 8 December 2010 1 Introduction There is a connection between the art of folding paper and mathematics. Paper folding is a beautiful and intricate art form

More information

Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees

Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Apex in a pyramid or cone, the vertex opposite the base; in

More information

Notes on finite group theory. Peter J. Cameron

Notes on finite group theory. Peter J. Cameron Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geometry (where groups describe in a very detailed way the

More information

Class One: Degree Sequences

Class One: Degree Sequences Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

More information

Intermediate Math Circles October 10, 2012 Geometry I: Angles

Intermediate Math Circles October 10, 2012 Geometry I: Angles Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,

More information

Proofs of the inscribed angle theorem

Proofs of the inscribed angle theorem I2GEO Proofs of the inscribed angle theorem University of South Bohemia Faculty of Education Jeronýmova 10 371 15 České Budějovice Czech Republic E-mail adress: leischne@pf.jcu.cz Abstract The inscribed

More information

Quadrilaterals Unit Review

Quadrilaterals Unit Review Name: Class: Date: Quadrilaterals Unit Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. ( points) In which polygon does the sum of the measures of

More information

4. How many integers between 2004 and 4002 are perfect squares?

4. How many integers between 2004 and 4002 are perfect squares? 5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

More information

1. An isosceles trapezoid does not have perpendicular diagonals, and a rectangle and a rhombus are both parallelograms.

1. An isosceles trapezoid does not have perpendicular diagonals, and a rectangle and a rhombus are both parallelograms. Quadrilaterals - Answers 1. A 2. C 3. A 4. C 5. C 6. B 7. B 8. B 9. B 10. C 11. D 12. B 13. A 14. C 15. D Quadrilaterals - Explanations 1. An isosceles trapezoid does not have perpendicular diagonals,

More information

Department of Pure Mathematics and Mathematical Statistics University of Cambridge GEOMETRY AND GROUPS

Department of Pure Mathematics and Mathematical Statistics University of Cambridge GEOMETRY AND GROUPS Department of Pure Mathematics and Mathematical Statistics University of Cambridge GEOMETRY AND GROUPS Albrecht Dürer s engraving Melencolia I (1514) Notes Michaelmas 2012 T. K. Carne. t.k.carne@dpmms.cam.ac.uk

More information

Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1) Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer

More information

College of Charleston Math Meet 2008 Written Test Level 1

College of Charleston Math Meet 2008 Written Test Level 1 College of Charleston Math Meet 2008 Written Test Level 1 1. Three equal fractions, such as 3/6=7/14=29/58, use all nine digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly one time. Using all digits exactly one

More information

DISCOVERING 3D SHAPES

DISCOVERING 3D SHAPES . DISCOVERING 3D SHAPES WORKSHEETS OCTOBER-DECEMBER 2009 1 . Worksheet 1. Cut out and stick the shapes. SHAPES WHICH ROLL SHAPES WHICH SLIDE 2 . Worksheet 2: COMPLETE THE CHARTS Sphere, triangle, prism,

More information

Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

More information

Chapter Three. Parallel Lines and Planes

Chapter Three. Parallel Lines and Planes Chapter Three Parallel Lines and Planes Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

More information

4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?

4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area? 1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of

More information

Geometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources:

Geometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources: Geometry Honors: Extending 2 Dimensions into 3 Dimensions Semester 2, Unit 5: Activity 30 Resources: SpringBoard- Geometry Online Resources: Geometry Springboard Text Unit Overview In this unit students

More information

SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.

More information

Year 10 Term 1 Homework

Year 10 Term 1 Homework Yimin Math Centre Year 10 Term 1 Homework Student Name: Grade: Date: Score: Table of contents 10 Year 10 Term 1 Week 10 Homework 1 10.1 Deductive geometry.................................... 1 10.1.1 Basic

More information

SHAPE, SPACE AND MEASURES

SHAPE, SPACE AND MEASURES SHAPE, SPACE AND MEASURES Pupils should be taught to: Use accurately the vocabulary, notation and labelling conventions for lines, angles and shapes; distinguish between conventions, facts, definitions

More information

3. If AC = 12, CD = 9 and BE = 3, find the area of trapezoid BCDE. (Mathcounts Handbooks)

3. If AC = 12, CD = 9 and BE = 3, find the area of trapezoid BCDE. (Mathcounts Handbooks) EXERCISES: Triangles 1 1. The perimeter of an equilateral triangle is units. How many units are in the length 27 of one side? (Mathcounts Competitions) 2. In the figure shown, AC = 4, CE = 5, DE = 3, and

More information

SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses

SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning

More information

VISUALISING SOLID SHAPES (A) Main Concepts and Results

VISUALISING SOLID SHAPES (A) Main Concepts and Results UNIT 6 VISUALISING SOLID SHAPES (A) Main Concepts and Results 3D shapes/objects are those which do not lie completely in a plane. 3D objects have different views from different positions. A solid is a

More information

IN SEARCH OF ELEMENTS FOR A COMPETENCE MODEL IN SOLID GEOMETRY TEACHING. ESTABLISHMENT OF RELATIONSHIPS

IN SEARCH OF ELEMENTS FOR A COMPETENCE MODEL IN SOLID GEOMETRY TEACHING. ESTABLISHMENT OF RELATIONSHIPS IN SEARCH OF ELEMENTS FOR A COMPETENCE MODEL IN SOLID GEOMETRY TEACHING. ESTABLISHMENT OF RELATIONSHIPS Edna González; Gregoria Guillén Departamento de Didáctica de la Matemática. Universitat de València.

More information

Name: 22K 14A 12T /48 MPM1D Unit 7 Review True/False (4K) Indicate whether the statement is true or false. Show your work

Name: 22K 14A 12T /48 MPM1D Unit 7 Review True/False (4K) Indicate whether the statement is true or false. Show your work Name: _ 22K 14A 12T /48 MPM1D Unit 7 Review True/False (4K) Indicate whether the statement is true or false. Show your work 1. An equilateral triangle always has three 60 interior angles. 2. A line segment

More information

Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid

Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 20, no. 1, pp. 101 114 (2016) DOI: 10.7155/jgaa.0086 Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid Yoshiaki Araki

More information

Creating Repeating Patterns with Color Symmetry

Creating Repeating Patterns with Color Symmetry Creating Repeating Patterns with Color Symmetry Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

More information

7.3 & 7.4 Polygon Formulas completed.notebook January 10, 2014

7.3 & 7.4 Polygon Formulas completed.notebook January 10, 2014 Chapter 7 Polygons Polygon 1. Closed Figure # of Sides Polygon Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 2. Straight sides/edges 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon 15 Pentadecagon

More information

A. 3y = -2x + 1. y = x + 3. y = x - 3. D. 2y = 3x + 3

A. 3y = -2x + 1. y = x + 3. y = x - 3. D. 2y = 3x + 3 Name: Geometry Regents Prep Spring 2010 Assignment 1. Which is an equation of the line that passes through the point (1, 4) and has a slope of 3? A. y = 3x + 4 B. y = x + 4 C. y = 3x - 1 D. y = 3x + 1

More information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

More information

Euler s Theorem. Tom Davis July 14, 2009

Euler s Theorem. Tom Davis  July 14, 2009 Euler s Theorem Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles July 14, 2009 Abstract Euler s theorem is a nice result that is easy to investigate with simple models from Euclidean

More information

Surface Area and Volume Nets to Prisms

Surface Area and Volume Nets to Prisms Surface Area and Volume Nets to Prisms Michael Fauteux Rosamaria Zapata CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version

More information

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: The point of intersection of the three medians of a triangle. Centroid Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

More information

Elements of Abstract Group Theory

Elements of Abstract Group Theory Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

More information

Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations

Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations Math Buddies -Grade 4 13-1 Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations Goal: Identify congruent and noncongruent figures Recognize the congruence of plane

More information

Page How many sides does an octagon have? a) 4 b) 5 c) 6 d) 8 e) A regular hexagon has lines of symmetry. a) 2 b) 3 c) 4 d) 5 e) 6 1 9

Page How many sides does an octagon have? a) 4 b) 5 c) 6 d) 8 e) A regular hexagon has lines of symmetry. a) 2 b) 3 c) 4 d) 5 e) 6 1 9 Acc. Geometery Name Polygon Review Per/Sec. Date Determine whether each of the following statements is always, sometimes, or never true. 1. A regular polygon is convex. 2. Two sides of a polygon are noncollinear.

More information

Surface Area and Volume Learn it, Live it, and Apply it! Elizabeth Denenberg

Surface Area and Volume Learn it, Live it, and Apply it! Elizabeth Denenberg Surface Area and Volume Learn it, Live it, and Apply it! Elizabeth Denenberg Objectives My unit for the Delaware Teacher s Institute is a 10 th grade mathematics unit that focuses on teaching students

More information

116 Chapter 6 Transformations and the Coordinate Plane

116 Chapter 6 Transformations and the Coordinate Plane 116 Chapter 6 Transformations and the Coordinate Plane Chapter 6-1 The Coordinates of a Point in a Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive

More information

Which two rectangles fit together, without overlapping, to make a square?

Which two rectangles fit together, without overlapping, to make a square? SHAPE level 4 questions 1. Here are six rectangles on a grid. A B C D E F Which two rectangles fit together, without overlapping, to make a square?... and... International School of Madrid 1 2. Emily has

More information

Lesson 10.1 Skills Practice

Lesson 10.1 Skills Practice Lesson 0. Skills Practice Name_Date Location, Location, Location! Line Relationships Vocabulary Write the term or terms from the box that best complete each statement. intersecting lines perpendicular

More information

MENSURATION. Definition

MENSURATION. Definition MENSURATION Definition 1. Mensuration : It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids. 2. Plane Mensuration : It deals with the sides, perimeters

More information

2014 Chapter Competition Solutions

2014 Chapter Competition Solutions 2014 Chapter Competition Solutions Are you wondering how we could have possibly thought that a Mathlete would be able to answer a particular Sprint Round problem without a calculator? Are you wondering

More information

15. Appendix 1: List of Definitions

15. Appendix 1: List of Definitions page 321 15. Appendix 1: List of Definitions Definition 1: Interpretation of an axiom system (page 12) Suppose that an axiom system consists of the following four things an undefined object of one type,

More information

Conjectures. Chapter 2. Chapter 3

Conjectures. Chapter 2. Chapter 3 Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

More information

The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons.

The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons. Interior Angles of Polygons The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons. The sum of the measures of the interior angles of a triangle

More information

Norman Do. Art Gallery Theorems. 1 Guarding an Art Gallery

Norman Do. Art Gallery Theorems. 1 Guarding an Art Gallery Norman Do Art Gallery Theorems 1 Guarding an Art Gallery Imagine that you are the owner of several art galleries and that you are in the process of hiring people to guard one of them. Unfortunately, with

More information