GGROUP THEORY AND MODERN DANCE COMPOSITION
ASHLEY WALLS WHITE
Abstract
This paper will examine the spatial reference systems typically used in Laban MovementAnalysis (LMA), and the consequences of group actions on these systems. The elementarynotions of inversion and transposition in choreographic composition can be defined in sucha way that they can be shown to be group homomorphisms in all the reference systemsof LMA. The notions of orbits and stabilizers on polyhedra are used to mathematicallydefine these choreographic devices, and these same notions can be used to define newchoreographic devices on the standard polyhedra used for spatial reference in dance.
Introduction
Laban Movement Analysis (LMA) was developed by Rudolf Laban as a means of describ-ing and documenting human movement, and is used as a tool for studying and analyzingmodern dance. The main aspects of LMA I will examine are the use of regular polyhedraas spatial reference systems, and the development of movement scales within these figure.The cube, octahedron, and icosahedron are used to determine 26 distinct spatial direc-tions, each coming from the vertices of these polyhedra. The notions of group actions,or symmetries, on these figures are surprisingly useful in modern dance choreography. Inchoreographic practice, as in music composition, a thematic phrase of movement materialis developed, and then choreographic devices are applied to this phrase to generate relatedand meaningful iterations of new movement material. One particular devices is the notionof inversion. Informally, inversion is defined as taking a movement to its spatial opposite,for instance inversion could take what was in the front space of the body to the backspace,or an action with an upright directional intent to having a downward directional intent.Inversion can be formally defined in the spatial reference systems, and we can show thatthis notion can be defined in terms of a group action, if we take the group to be the verticesof any of the three noted polyhedrons.
Inversion
The Octahedron.
We will start by examining the octahedron. We can consider the bodysituated inside an octahedron. Let v be the head, v be the feet, and v be the front of thebody so that v is the middle side left direction and v is the middle side right. We defineinversion as taking the directional intent of an action to its spatial opposite. Using the Date : May 23, 2020. a r X i v : . [ m a t h . HO ] M a y ASHLEY WALLS WHITE vertices of the octahedron, we would hope that the inversion of an action in the directionof v would be an action in the direction of v , and similarly the inversion of an object isthe direction of v would be the same action in the direction v . Figure 1.
OctahedronFormally consider the direction of an action to be the set containing the vertex thatdefines a particular direction, so an action that takes place in the middle right directionwould be in the direction { v } .Consider the set of the vertices V = { v , v , v , v , v , v } and the group of rotationalsymmetries on the octahedron. Considering { v i } for i = 5 ,
6, the stabilizer of v i is theset of rotations that leave v i fixed. The stabilizer H = stab ( v i ) fixes both v and v andinduces a group action on the octahedron. There is only one permutation other than theidentity in the stabilizer of v i . H = stab ( v i ) = { e, ( v , v )( v , v ) } , for i = 5 , ROUP THEORY AND MODERN DANCE COMPOSITION 3 orb ( v ) = { v , v } orb ( v ) = { v , v } Since the orbits partition the set V , for an action in any direction, we can define theinversion of that action to be the only other element in the action’s orbit. Considering theaction in the middle back direction, { v } , we can define the inversion I ( v ) = orb ( v ) − { v } = { v } which gives us an action in { v } , the middle front direction, which is what we expectedinversion to be based on our informal definition. Similarly, I ( v ) = orb ( v ) − { v } = { v } I ( v ) = orb ( v ) − { v } = { v } I ( v ) = orb ( v ) − { v } = { v } Actions in the mid level directly left and directly right are inversions of each other, andtaking the stabilizer of any other vertex in the octahedron puts these directions in the sameorbit.In general, for an invertible action { v i } , define I ( v i ) = orb ( v i ) − { v i } Inversion in the octahedron sets the stage for defining choreographic devices on the icosa-hedron, which will provide a richer description of directions in space. In the icosahedronwe will continue the practice of examining orbits and stabilizers to define inversion, but aclearer distinction will be made between group actions that define each type of inversion,high-low, front-back, and left right.
The Icosahedron.
For our purposes it will be most convenient to consider the vertices ofthe icosahedron formed by the corners of the mutually orthogonal intersecting rectangles.In anatomy we consider the vertical, horizontal, and sagittal planes on the body. Thevertical plane cuts the body into front and back halves, the horizontal plane cuts the bodyinto top and bottom halves, and the sagittal plane cuts the body into left and right halves.[1]In the figure above, the blue plane is the sagittal plane, the green plane is the horizontalplane, and the red plane is the vertical plane. If we situate the body inside the icosahedronso that the head is at the top of the sagittal plane, the vertices correlate to the followingdirections of the body in space.
ASHLEY WALLS WHITE
Figure 2.
Icosahedron from Orthogonal Rectangles v - forward high v - middle right back v - back high v - middle left back v - high right v - forward low v - high left v - back low v - middle right forward v - low right v - middle left forward v - low leftLet’s first begin to consider the group of rotational symmetries on the figure, I r . Weonly want symmetries that will keep the same orientation of the head and feet in the sagit-tal plane. Allowing I r to act on the edges of the icosahedron, note that keeping the ”head”in place might correlate to taking the stabilizer of the top edge connecting v and v , callit e . Considering stab ( e
1) we find that this is just the rotational symmetry ρ .The rotational symmetry ρ is working about a 2-fold axis that passes through the mid-point of e to the midpoint of the opposite edge, the edge connecting v and v . Thissymmetry induces a group action on the vertices, from which we can then determine thepartition of the vertices into orbits. ROUP THEORY AND MODERN DANCE COMPOSITION 5 orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } From choreographic composition, there are many ways to interpret inversion. The threewe will focus on in this section are from forward to back, from low to high and from left toright. For the front-back inversion, the inversion of forward high, v , is back high, v , andfrom above those two vertices are in the same orbit. All the orbits defined above partitionvertices into pairs with their front-back inversion, I fb , except for the permutations ( v , v )and ( v , v ). These two orbits give define a relation between left and right directions,which does not fit into the definition of a front-back inversion, so we need an action on thevertices that will fix this entire plane, i.e. the vertices v , v , v , v . Considering the stab ( v i , v i +1 ), for i = 3 ,
11 we arrive at the following: H = stab ( v i , v i +1 ) = { e, ( v , v )( v , v )( v , v )( v , v ) } The non-identity permutation defines the following orbits: orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } This partitions the orbits into sets containing an action and that action’s front-back in-version. Another symmetry will have to be determined to define low-high inversion andleft-right inversion.
Low-High Inversion.
For low-high inversion, I lh , we again cannot employ a rotationalsymmetry. First consider again ρ , but consider it about the axis that goes through thehorizontal place, entering at the edge connecting v and v , and existing the edge connect-ing v and v . The partition of the vertices into orbits under this action is as follows: orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } ASHLEY WALLS WHITE orb ( v ) = { v = e, v } This works except for all the vertices in the horizontal plane. These do not have low-high inversions, so we need an action on the vertices that will fix this entire plane, i.e. thevertices v , v , v , v . Considering the stab ( v i , v i +1 ), for i = 5 .. H = stab ( v i , v i +1 ) = { e, ( v , v )( v , v )( v , v )( v , v ) } The non-identity permutation defines the following orbits: orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } which correctly determines the sets of elements with their low-high inversion. Left-Right Inversion.
Left-Right inversion is similarly defined by taking the stabilizerof sagittal plane.Considering the stab ( v i , v i +1 ), for i = 1 .. H = stab ( v i , v i +1 ) = { e, ( v , v )( v , v )( v , v )( v , v ) } The non-identity permutation defines the following orbits: orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } orb ( v ) = { v = e, v } We find that what is referred to as ”inversion” in modern dance choreography followsa reflective symmetry in the icosahedron.1.
Additional Uses for Group Actions
Subsequent inversions on the icosahedron.
In choreography an inversion can be ap-plied to an action multiple times in a row. Consider front-back inversion on the icosahedronand actions A and B , where action A = { v } is in the direction of v and action B = { v } is in the direction of v . ROUP THEORY AND MODERN DANCE COMPOSITION 7 I fb ( A ) = { v } I fb ( B ) = { v } Define the operation P ⊕ R on actions P and R to be performing an action in the di-rection of P, and then performing an action in the direction R , the resulting action justbeing in the direction R . Then, I fb ( A ) ⊕ I fb ( B ) = { v } .And A ⊕ B = { v } ,so that I fb ( A ⊕ B ) = { v } .Performing two actions in sequence and then taking the inversion yields the same resultsas doing the inversions of each of the actions in sequence. We can define the inversion of asequence of events to be sequence of each individual action’s inversion. It should be notedthat this definition is only concerned with the end destination of the movement, but thenot the sequence of movements taken to get there. Laban’s normal zones as group actions.
Laban identified five principle zones of thebody associated with the head, each arm, and each leg. [3] Considering the body positionedin the icosahedron as before, consider the right and left arms at high right { v } and highleft { v } respectively, with the legs similarly at { v } and { v } . If we take the stabilizerof any of these vertices we can create orbits which roughly correspond to Laban’s normalzones for the arms and legs. Consider first stab ( v ). Under this the following orbits arecreated. orb ( v ) = { v , v , v , v , v } orb ( v ) = { v , v , v , v , v } orb ( v ) = { v } orb ( v ) = { v } The first, orb ( v ) gives a group of vertices within a normal range for the left arm, with v being the standard position for the left arm. The normal range, which relates to Laban’snormal zones, for any arm or leg can be defined as the orbit of any of the vertices adja-cent to the vertex that gives the standard position for that limb under the stabilizer of thevertex that gives the standard position. Thus the normal zone for each limb is the following: ASHLEY WALLS WHITE
Limb Standard Position Normal RangeLeft Arm { v } orb ( v ) = { v , v , v , v , v } Right Arm { v } orb ( v ) = { v , v , v , v , v } Left Leg { v } orb ( v ) = { v , v , v , v , v } Right Leg { v } orb ( v ) = { v , v , v , v , v } Laban’s Directions in Space
Laban gave a codified system for representing directions in space. Henceforth we willrefer to the directions using his symbols, which are listed below [3].
Figure 3.
Laban’s Directions in SpaceBelow is the icosahedron with these directions labeled.Using these spatial directions Laban established what he felt were natural sequences ofmovement inside the icosahedron and other spatial systems. He claimed these sequencesfollowed natural patterns of the body and were governed by scientific and mathematical
ROUP THEORY AND MODERN DANCE COMPOSITION 9
Figure 4.
Directions on the Icosahedronprinciples. Laban named many of these trace forms, including the defense and attackscales, and the girdle. We will start the exploration of the principles governing these scaleson the most intriguing trace form, the primary scale on the icosahedron.
The Primary Scale on the Icosahedron
Laban established what he called the standard (or primary) scale on the twelve verticesof the icosahedron, and it is displayed below. Laban recognized many symmetries andpatterns used to establish this scale, including squares, triangles and diagonals in space.
Figure 5.
Laban’s Primary Scale
ROUP THEORY AND MODERN DANCE COMPOSITION 11
Louis Horst was one of the first to recognize a possible connection between music com-position and dance composition, and the devices of inversion and transposition in dancewere translated from these same devices in music theory [2]. Taking a hint from musictheory, in an attempt to better understand Laban’s reasoning behind the primary scale,we can visualize the primary scale on a clock, akin to the musical clock from music theory[4].
Figure 6.
Primary Scale ClockThis illustrates how the primary scale can be represented by Z . And beautifully,the triangles, quadrangles and diameters referred to by Laban can be viewed as cosets of Z . The triangles correlate to 4 Z , the quadrangles 3 Z and the diameters 6 Z . WhatLaban referred to as trace forms can be represented by paths connecting points of the clock. Figure 7.
Triangles on the Clock
ROUP THEORY AND MODERN DANCE COMPOSITION 13
Figure 8.
Squares on the Clock
Defining Inversion on the Primary Scale Clock
Earlier we defined three different types of inversion in the icosahedron; front-back, left-right, and high low. We can now define another type of inversion, diametral inversion, onthe primary scale clock. Diametral inversion could be defined as subsequent applicationsof the previously defined inversion. For instance the diametral inversion of an action inthe right middle front direction would be an action in the left middle back direction, whichcould be obtained by applying a front back inversion, and then a left right inversion. Thisprevious analysis focuses only on the final destination of the movement, rather than thepath taken to arrive there, so we were only able to determine the inversion of a singlemovement, rather than the inversion of a sequence of movements. If we now refer onlyto diametral inversion, we can develop a method of determining the inversion for a wholesequence of movements using the primary scale clock.
Figure 9.
Diameters Defining Diametral Inversion
ROUP THEORY AND MODERN DANCE COMPOSITION 15
Figure 10.
The ”Girdle” Trace Form and It’s Inversion
New Devices
The goal of this paper was to first mathematically establish the symmetries that de-fine choreographic devices already in place, with the hope of then exploring additionalsymmetries to determine new devices. We have currently only looked at the concept ofchoreographic inversion, and shown how it can be determined by group actions on theoctahedron and icosahedron. More analysis can be done on these polyhedra, and in thefuture to determine new devices it may be useful to examine non-regular and non-convexpolyhedra as systems of spatial reference.
References [1] I. Bartenieff and D. Lewis
Body Movement
Modern Dance Forms
Choreutics
Prelude to Musical Geometry , The College Mathematics Journal29