Ratio Mathematica ISSN: 1592-7415 Vol. 31, 2016, pp. 93--110 eISSN: 2282-8214 93 Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube Ferdinando Casolaro1, Luca Cirillo2, Raffaele Prosperi3 1Department di Architecture University “Federico II” of Naples, Italy ferdinando.casolaro@unina.it 2 University of Sannio in Benevento, Italy luca.cirillo@unisannio.it 3 DISUFF University of Salerno, Italy rprosperi@unisa.it Received on: 3-11-2016. Accepted on: 11-11-2017. Published on: 28-02-2017 doi: 10.23755/rm.v31i0.322 © Casolaro et al. Abstract This paper deals with groups of transformations with finite number of isometries and extends previous studies (Casolaro, F. L. Cirillo and R. Prosperi 2015) which are related to endless groups of transformations with isometrics. In particular, isometries of the tetrahedron and cube, which turn these figures in itself, are presented. Keywords: Geometric transformations, isometries, symmetry. 2010 AMS subject classification: 97G50; 51N25. Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 94 1. Introduction Compared with the operation of product of isometries, in previous studies, we presented some examples of infinite groups of transformations, whose we highlighted the following properties: - The isometries of the space form a group. - The direct isometries of the space form a group, subgroup of the previous group. - The translations of the space form a group, subgroup of the group of direct isometries. - Rotations around a straight form a group, subgroup of direct isometries. - The helical movements all having the same axis form a group, subgroup of the group of direct isometries. In this case, since the helical movements turn out to be products of rotations for translations having the direction of the axis of rotation, also translations (the rotation is reduced to the identity) and rotations (the translation is reduced to the identity) may be considered helical movements. It is also possible to obtain groups of transformation with a finite number of isometries. In particular: about the tetrahedron, we show the axial symmetry μ having as an axis line r, rotations ρ of 120 ° and 240 ° around the height of the tetrahedron outgoing from a fixed vertex, planar symmetry σ relative to the plan π passing through two vertices of the tetrahedron and through the midpoint of the edge that joins the other two vertices; about the cube, rotations ρ around a line r connecting the centers of two opposite faces, rotations ρ around the line r joining the midpoints of two opposite edges, planar symmetry σ relative to the plan π passing through two vertices of the tetrahedron and through the midpoint of the edge that joins the other two vertices, planar symmetry σ relative to the pane π parallel to two faces passing through the midpoints of the four edges perpendicular to these two faces, planar symmetries σ relative to the pane π passing through two opposite edges that do not have face in common and a vertex in common. Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube 95 Figure 1 Consider three straight lines x, y, z, passing through the same point O and perpendicular to each other two by two. The three planes α, β, γ, respectively determined by the straight lines x and y, x and z, and y and z, are also perpendicular to each other two by two (Figure 1). Let be: I the identity, sx the axial symmetry having as an axis the line x, sy the axial symmetry having as an axis the line y, sz the axial symmetry having as an axis the line z, sα the planar symmetry relative to the plane α, sβ the planar symmetry relative to the plane β, sγ the planar symmetry relative to the plane γ, so the symmetry with center O, It occurs that these eight isometries form a group. For this purpose, it is sufficient to prove that the product of any two of them is still one of the eight indicated isometries. 2. Tetrahedron’s Isometries Other examples of finite groups of isometries can be obtained considering all the isometries which leave fixed a given figure F, that is, such that in each of them F is united (F is transformed into itself). ABCD and A'B'C'D 'are two congruent tetrahedra. Then there exists one and only one isometry that transforms the vertices A, B, C, D neatly in the vertices A ', B', C ', D' (Figure 2). This isometry is direct or reverse depending on whether or not the two tetrahedra are equally oriented. Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 96 Figure 2 Isometries that turn a tetrahedron T into itself are 24 (twenty-four). They form a group ST , obviously isomorphic to the group S4 of the 24 permutations on four letters A, B, C, D. Among the isometries ϕ that transform the tetrahedron T into itself, we present the following: a) The axial symmetry μ having as an axis the straight line r, joining the midpoints of two opposite sides (bimedian), is a rotation of 180 ° around the straight line r. The symmetries of this type present in the group are 3 (as many as the pairs of opposite sides of the tetrahedron); they have evidently period 2. Therefore there are 3 axial symmetries that leave T globally invariant, as many as the pairs of opposite sides. A substitution is associated with each of these symmetries (M. Impedovo 1998). - With symmetry μ1 about the line r1 joining the midpoints of the sides AB and CD, the following substitution is associated: - With symmetry μ2 about the line r2 joining the midpoints of the sides AC e BD the following substitution is associated: - With symmetry μ3 about the line r3 joining the midpoints of the sides AD e BC the following substitution is associated: Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube 97 b) The rotations ρ of 120 ° and 240 ° around the height of the tetrahedron outgoing from a fixed vertex. For each height of the tetrahedron, you have two rotations of period 3 which hold the summit fixed. Since the tetrahedron heights are 4, these rotations are 8; therefore, there are 8 rotations of this type which transform T into itself, two for each height of the tetrahedron. A substitution is associated with each of these rotations. - With rotation ρ1 about the height outgoing from A the following substitution is associated: relative to the amplitude of 120° relative to the amplitude of 240° - With rotation ρ3 about the height outgoing from B the following substitution is associated: relative to the amplitude of 120° relative to the amplitude of 240° - With rotation ρ5 about the height outgoing from C the following substitution is associated: relative to the amplitude of 120° relative to the amplitude of 240° - With rotation ρ7 about the height outgoing from D the following substitution is associated: Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 98 relative to the amplitude of 120° relative to the amplitude of 240° c) The planar symmetry σ relative to the plan π passing through the two vertices of the tetrahedron and the midpoint of the edge that joins the other two vertices. The σ symmetry σ is uniquely determined by the initial vertex. The symmetries of this type are 6 (as many as the pairs of vertices of the tetrahedron), and have period 2. A substitution is associated with each of these symmetries - With symmetry about the plane ABM1, with M1 medium point of CD, the following substitution is associated: - With symmetry about the plane ACM2, with M2 medium point of BD, the following substitution is associated: - With symmetry about the plane ADM3, with M3 medium point of BC, the following substitution is associated: - With symmetry about the plane BCM4, with M4 medium point of AD, the following substitution is associated: Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube 99 - With symmetry about the plane BDM5, with M5 medium point of AC, the following substitution is associated: - With symmetry about the plane CDM6, with M6 medium point of AB, the following substitution is associated: It is observed that the two sets of isometries described in points a) and b) each supplemented with the identity are closed about to the product. The first set is a G1 group of order 4 of involutorie transformations. The second set is a G2 group of order 9 of periodic transformations of order 3. The union of the two groups is a G3 group of order 12, which is the group of direct isometries of T. We will now examine the product of three symmetries, or we will fix an isometry σk of type c) (planar symmetry), and we will consider an isometry αt (t = 1, 2, … , 12) variable in the G3 group. The product σk º αt is still an isometry that changes the tetrahedron T into itself. They are in number of 12; in fact, if we fix, for example, the isometry multiplying each isometry of the G3 Group for σ1, we will get 12 reverse isometries reverse, which can be summarized as: Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 100 It is easily seen that it results: ϕ 12 = σ 1 , ϕ 5 = σ 2 , ϕ 4 = σ 3 , ϕ 7 = σ 4 , ϕ 6 = σ 5 , ϕ 1 = σ 6 That is the 12 isometries σk º αt are given by the 6 planar symmetries σk of the type c) and by the 6 antirotations ϕ k, with period 4. The isometries ϕ k do not take firm no vertex and no edge of the tetrahedron. In summary, we can say that the three axial symmetries of the G1 group, the 8 rotations of the G2 group, the 6 planar symmetries, the 6 latest found isometries, along with the identity, are the 24 isometries that leave the tetrahedron T globally invariant; their set is the ST group of isometries of T. ST is the group of isometries that change the tetrahedron T in itself. Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube 101 3. Isometries of Cube Some examples of finite groups of isometries can be had considering all isometries leaving globally invariant a cube (A. Morelli, 1989). ABCDEFGH e A’B’C’D’E’F’G’H’ are two equal cubes. Then there exists one and only one isometry that transforms the vertices A, B, C, D, E, F, G, H, neatly in the vertices A’, B’, C’, D’, E’, F’, G’, H’ (Figure 3). This isometry is direct or reverse depending on whether or not the two cubes are equally oriented. Figure 3 Isometries that transform a C cube to itself are forty eight. They forming a Sc group evidently isomorphic to S8 group of forty eight permutations on eight letters A, B, C, D, E, F, G, H. Among the isometries that transform the C Cube itself there are obviously the following: a) The rotations around a straight line r which joins the centers of two opposite faces. Since the faces of the cube are six, these lines are three; for each of these straight lines the cube is transformed into itself by the amplitude rotations, respectively, 90°, 180°, 270°. Therefore you have nine rotations of this type which transform C itself. For each of these rotations it is associated a substitution. - To 1 rotation around the straight through M1M2, with M1 the center of the ABCD face and M2 the center of the EFGH face, is associated the substitution:       EHGFCBAD HGFEDCBA : 1  relative to the amplitude of 90° Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 102       FEHGBADC HGFEDCBA : 2  relative to the amplitude of 180°       GFEHADCB HGFEDCBA : 3  relative to the amplitude of 270° - To 4 rotation around the straight through M3M4, with M3 the center of the ABFE face and M4 the center of the DCGH face, is associated the substitution:       AHEDCFGB HGFEDCBA : 4  relative to the amplitude of 90°       BADCFEHG HGFEDCBA : 5  relative to the amplitude of 180°       GBCFEDAH HGFEDCBA : 6  relative to the amplitude of 270° - To 7 rotation around the straight through M5M6, with M5 the center of the AEHD face and M6 the center of the BFGC face, is associated the substitution:       EFCDABGH HGFEDCBA : 7  relative to the amplitude of 90°       DCBAHGFE HGFEDCBA : 8  relative to the amplitude of 180°       ABGHEFCD HGFEDCBA : 9  relative to the amplitude of 270° b) The rotations  around the straight line r that connects the midpoints of two opposite edges. Since the edges of the cube are twelve, these lines are six; for Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube 103 each of these straight lines the cube is transformed into itself by rotations of 180 ° amplitude. For each of these rotations it is associated a substitution. - To rotation 10 around the straight line joining the midpoints of AB and EF edges, is associated the substitution: -       DCFEHGBA HGFEDCBA : 1 0  - To rotation 11 around the straight line joining the midpoints of CD and HG edges, is associated the substitution:       HGBADCFE HGFEDCBA : 1 1  - To rotation 12 around the straight line joining the midpoints of BC and HE edges, is associated the substitution: -       HADEFCBG HGFEDCBA : 1 2  - To rotation 13 around the straight line joining the midpoints of AD and FG edges, is associated the substitution:       BGFCDEHA HGFEDCBA : 1 3  - To rotation 14 around the straight line joining the midpoints of BC and HE edges, is associated the substitution: -       DADEFCBG HGFEDCBA : 1 4  - To rotation 15 around the straight line joining the midpoints of AD and FG edges, is associated the substitution: Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 104         BGFCDEHA HGFEDCBA :1 5 c) The rotations  around the straight line r that contains a diagonal. The number of se lines is four; for each of these straight lines the cube is transformed into itself by the amplitude rotations respectively 120° and 240°. Therefore there are eight rotations of this type which transform C to itself. For each of these rotations it is associated a substitution. - To rotation 16 around the diagonal AF, it is associated the substitution:       FCDEHABG HGFEDCBA : 1 8  relative to the amplitude of 120°       DAHEFGBC HGFEDCBA : 1 9  relative to the amplitude of 240° - To rotation 18 around the diagonal BE, it is associated the substitution:       FCDEHABG HGFEDCBA : 1 8  relative to the amplitude of 120°       DAHEFGBC HGFEDCBA : 1 9  relative to the amplitude of 240° - To rotation 20 around the diagonal CH, it is associated the substitution:       HEDABCFG HGFEDCBA : 2 0  relative to the amplitude of 120°       HABGFCDE HGFEDCBA : 2 1  relative to the amplitude of 240° - To rotation 21 around the diagonal DG, it is associated the substitution: Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube 105       BGHADEFC HGFEDCBA : 2 2  relative to the amplitude of 120° -       FGBCDAHE HGFEDCBA : 2 3  relative to the amplitude of 240° d) The planar symmetry with respect to  plane parallel to two faces through the midpoints of the four edges perpendicular to these two faces. The symmetries of the type indicated are three. For each of these symmetries it is associated a substitution. - At the planar symmetry 1 with respect to the plane 1 parallel to ABGH and EFCD faces, is associated the substitution: -       EFGHABCD HGFEDCBA : 1  - At the planar symmetry 2 with respect to the plane 2 parallel to ABDC and HGEF faces, is associated the substitution:         ABCDEFGH HGFEDCBA :2 - At the planar symmetry  with respect to the plane 3 parallel to BCGH and ADHE faces, is associated the substitution: -         ABCDEFGH HGFEDCBA :3 e) The symmetries with respect to the  plan through two opposite edges that do not have common face and vertex. The symmetries of the type indicated are six. For each of these symmetries it is associated a substitution. - At the planar symmetry 4 respect to the 4 plan through the edges AD and GF is associated with the substitution: Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 106 -       BGFCDEHA HGFEDCBA : 4  - At the planar symmetry 5 respect to the 5 plan through the edges BC and HE is associated with the substitution: -       HADEFCBG HGFEDCBA : 5  - At the planar symmetry 6 respect to the 6 plan through the edges AB and EF is associated with the substitution:       DCFEHGBA HGFEDCBA : 6  - At the planar symmetry 7 with respect to the 7 plan through the edges CD and HG is associated with the substitution: -       HGBADCFE HGFEDCBA : 7  - At the planar symmetry 8 with respect to the 8 plan through the edges AH and CF is associated with the substitution:       HEFGBCDA HGFEDCBA : 8  - At the planar symmetry 9 with respect to the 9 plan through the edges BG and DE is associated with the substitution:       FGHEDABC HGFEDCBA : 9  Note that the two sets of isometry described in points a), b) and c), each supplemented with the identity: Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube 107       HGFEDCBA HGFEDCBA I : , are closed with respect to the product. The first set G1 is a group of order ten, the second set is a group G2 of order seven, the third set is a group G3 of order nine. The union of these three groups is a G4 group of order twenty four which constitutes the group of direct isometries of C. Let us now examine the product of three symmetries, that is fix an type d) isometry k (planar symmetry), and consider an isometry t (t = 1, 2, ..., 24) variable in the G4 group. The product k t is still an isometry which changes the C Cube to itself. The number of these product is twenty four; in fact, it fixed eg. the isometry       EFGHABCD HGFEDCBA : 1  , multiplying each isometry of the G4 group , you get twentyfour reverse isometries, which can be summarized as:        FGHEDABC HGFEDCBA : 111   ,        GHEFCDAB HGFEDCBA : 221   ,        HEFGBCDA HGFEDCBA : 331   ,        DEHABGFC HGFEDCBA : 441   ,        CDABGHEF HGFEDCBA : 551   ,        FCBGHADE HGFEDCBA : 661   , Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 108        DCFEHGBA HGFEDCBA : 771   ,        ABCDEFGH HGFEDCBA : 881   ,        HGBADCFE HGFEDCBA : 991   ,        EFCDABGH HGFEDCBA : 1 01 01   ,        EBGHEFCD HGFEDCBA : 1 11 11   ,        EDAHGBCF HGFEDCBA : 1 21 21   ,        CFGBAHED HGFEDCBA : 1 31 31   ,        EDAHGBCF HGFEDCBA : 1 41 41   ,        CFGBAHED HGFEDCBA : 1 51 51   ,        GFCBADEH HGFEDCBA : 1 61 61   ,        CFEDAHGB HGFEDCBA : 1 71 71   ,        EDCFGBAH HGFEDCBA : 1 81 81   , Groups of Transformations with a Finite Number of Isometries: the Cases of Tetrahedron and Cube 109        EHADCBGF HGFEDCBA : 1 91 91   ,        ADEHGFCB HGFEDCBA : 2 02 01   ,        GBAHEDCF HGFEDCBA : 2 12 11   ,        AHGBCFED HGFEDCBA : 2 22 21   ,        CBGFEHAD HGFEDCBA : 2 32 31   ,        EFGHABCD HGFEDCBA I : 2 41   . It is easily seen that results:  24 =  1 ,  8 =  2 ,  2 =  3 ,  7 =  6 ,  9 = 7 ,  3 =  8 ,  1 =  9 that is, the twentyfour isometries k t are given from nine symmetries k planar type d), e), and fifteen anti rotations k. In summary therefore it can be said that the twenty three rotations of the G4 group, the nine planar symmetries and the latest isometries found, along with the identity, are the forty eight isometries which leave the cube C globally invariant; their set is the SC group of isometries of the cube C. SC is the group of the isometries that change C cube to itself. Conclusions As already shown in a previous work (Casolaro, F., Cirillo, L. and Prosperi, R. 2015), the geometric Universe is three-dimensional, so the transformations taking place in it are generated in space. Then, we believe, for a correct analysis of the physical phenomena that occur in the universe, that it is essential to the knowledge of the real transformations that take place in it. Recent results of other branches of mathematics, in particular the modern algebra, have Ferdinando Casolaro, Luca Cirillo, Raffaele Prosperi 110 highlighted the interrelationships between movements in the plane and in space with some properties of the Theory of Groups (Casolaro, F. 1992), for which we consider essential to the deepening of these issues both in education and in the field of pure research (Casolaro, F. and Eugeni, F. 1996). Unfortunately, teaching (Casolaro F. 2014) in both the Secondary School that the University has been anchored to old programs that do not take into account the development of mathematics in the last 150 years, so we hope that this work will stimulate teachers and researchers to expand their views. References Casolaro, F. L. Cirillo and R. Prosperi (2015). “Le Trasformazioni Geometriche nello Spazio: Isometrie”. Science &Philosophy n. 1, 2015. Casolaro, F. and Eugeni, F. (1996). “Trasformazioni geometriche che conservano la norma nelle algebra reali doppie”. Ratio Matematica n. 1, 1996. Casolaro, F. and Prosperi, R. (2011). “La Matematica per la Scuola Secondaria di secondo grado: un contributo per il docente di Matematica". Atti della “Scuola estiva Mathesis” 26-30 luglio 2011. Terni: Editore 2C Contact. Casolaro, F. and Paladino L. (2012). “Evolution of the geometry through the Arts”. 11th International Conference APLIMAT 2012 in the Faculty of Mechanical Engineering - Slovak University of Tecnology in Bratislava, febbraio 2012. Casolaro, F. (2014). “L'evoluzione della geometria negli ultimi 150 anni ha modificato la nostra cultura. Lo sa la Scuola?”. “Science&Philosophy Journal of Epistemology", Volume 2, Numero 1, 2014. Cundari, C. (1992). “Disegno e Matematica per una didattica finalizzata alle nuove tecnologie”. Progetto del M.P.I. e del Dipartimento di Progettazione e Rilievo dell’Università “La Sapienza” di Roma, 11-15 dicembre 1990; 6-10 maggio 1991; 8-12 dicembre 1991. Casolaro, F. (1992). “Gruppo delle Affinità, Gruppo delle Similitudini, Gruppo delle Isometrie”. Da C. Cundari, Progetto del M.P.I. e del Dipartimento di Progettazione e Rilievo dell’Università “La Sapienza” di Roma (1992). Morelli, A. (1989). Geometria per il biennio delle scuole medie superiori. Napoli: Edizione Loffredo. Impedovo, M. (1998). “Matrici e isometrie nello spazio”. Un utilizzo didattico di MAPLE. L’insegnamento della matematica e delle scienze integrate, vol. 21 B, n° 1, febbraio 1998.