Graphics Calculator Applications to Maximum and Minimum Problems on Geometric Constructs

Transcription

1 Graphics Calculator Applications to Maximum an Minimum Problems on Geometric Constructs Expressing the size of constructs in terms of the istances between the vertices enables one to easily examine other interesting ieas beyon just the size of a construct (point, line segment, trigon, tetraheron, pentatope,) This paper will focus upon the variance, if any, of various attributes of a construct when the istance between a pair of vertices varies The chosen construct is a -D tetraheron since this can be "really" moelle an allows for iscussions involving -D, -D, -D, an 0-D constructs, which can also be reaily perceive Further, to a a little spice to the iscussion, -D angle an -D angle variations will be examine Although a general formulation for the size of any construct will be exploite, for the purposes of gathering specific ata a special tetraheron has been chosen This particular tetraheron is sort of unique in that it is compose of the smallest six consecutive integral lengths that etermine a tetraheron with integral volume This problem was pose in the Mathematics Magazine, February, 97, Problem b an solve by this primary author using the prementione general formula This general formula was originally formulate an proven for the (0-)-D constructs by this primary author uring November, 99, an extene to n-d constructs The extension was proven by Dr Eugene Curtin an this author uring the Fall of 994 Also, a special -D angular ruler, accurate to 0 - raians, was create to irectly measure -D angles, an L Huilier s formula was moifie to express -D angles in steraians The seconary author formatte the calculators an computers, which greatly enhance response time an cognitive unerstaning with graphical impact An, transferre this paper an technological formatings to be publishe in the electronic proceeings The general formula for counting the cubes (point, line segment, square, cube, tesseract,) of a space that tessellate a construct of that space in terms of the istances between the vertices of the construct is where mc ( ) D nxn n n! D nxn n n nn u base upon the istances between three vertices, V o, V i as follows an u n enotes a unit cube of the n-d space that countable tessellates the n-d space Choose a vertex, V o, of the construct V i an V j are any other vertices of the construct (V o ) represents the istance between vertex V o an vertex V j n(v o, V i ) represents the numerical part of that istance between V o an V i n, ij is

2 Similarly, ij n(v o ) an n(v i ) represent corresponing numerical parts [ n( V, V + n( V, V ) ji [ ] [ n( V V 0 i o j i, j, see the first iagram Special case, n V V kk Diagram A representation of the components of ij [ ( [ ( [ ( 0, k + n V0, Vk n Vk, Vk [ nv ( 0, Vk) ] + [ nv ( 0, Vk) ] 0 [ nv ( V 0, k, epicte in Diagram Diagram A representation of the components of special case kk -D Attributes The measure of a -D construct, the volume of a tetraheron, as etermine by the lengths of the eges is: m ( V ) D u (, b, c,, e, f ) a! x

3 u since + e c + f b + e c e e + f a + f b e + f a f [ n( V0, V [ n( V0, V + [ n( V0, V [ n( V, V [ n( V0, V + [ n( V0, V [ n( V, V [ n( V0, V e [ n( V0, V) ] + [ n( V0, V [ n( V, V [ n( V, V f 0 e + e + f + f c b a u, as epicte in Diagram

4 Diagram A general tetraheron To calculate the volume of any tetraheron, program your calculator to calculate: + e c + f b + e c e e + f a + f b e + f a f an to ask for "a,b,c,,e,f " as input ata The following TI- program accomplishes the calculation of the volume: :Disp "THIS PROGRAM WILL FIND THE VOLUME OF ANY TETRAHEDRON" :Prompt A,B,C,D,E :*D L :D+E -C M :D +F -B N :D +E -C O :*E P :E +F -A Q :D +F -B R :E +F -A S :*F T :(/) ((/)*et [[L,M,N][O,P,Q][R,S,T]]) V :Disp "THE VOLUME IS:" :Disp V The tetraheron chosen as a basis for specific ata is the one with the smallest six consecutive integral lengths that etermine an integral volume as epicte in Diagram 4:

5 Diagram 4 Tetraheron with smallest consecutive integral lengths an integral volume Check your calculator to etermine, ( V 7 ) 4u m (,9,,0,,) For the purposes of gathering ata without loss of generality, an iscussing variable attributes of this specific trigon, we will allow ege (V 0, V ) to vary, ie f will be ientifie as a variable, x Program a function, f, such that f () x + e c b e + e e c a e x a b an to ask for "a,b,c,,e" The following TI- program accomplishes the calculation of the volume as a function of x : :Disp "THIS PROGRAM WILL FIND THE VOLUME OF ANY TETRAHEDRON FOR VALUES OF X" :Prompt A,B,C,D,E :"(/) ((/)((D (E *X -(E +X -A ) ))-(D +E -C )((D +E -C )X -(E +X -A )(D +X -B ))+(D +X - B )((D +E -C )(E +X -A )-E (D +X -B ))))" Y :0 Xmin

6 : Xmax :0 Ymin :0 Ymax : Xscl :0 Yscl :DispGraph The graphics calculator will isplay the graph of f an can isplay omain an range values of f The graph combine with orere pairs allows one to approximate the omain of f, ie the minimum an maximum values of x an to observe an approximate functional optimal values of f(x), ie the minimum volume, 0 u, an the maximum volume The following iscrete subsets of f give the functional values of f at integral values of x, subset, an a process of refining the minimum an maximum values of x, subset, subset Subset of f escribes the volume of the given tetraheron (to the nearest 0 - u ) as a function of integral istances between V 0 an V S {(, 79), (4, 74), (5, 7777), (, 45), (7, 4), (, 507), (9, 5599), (0, 57), (, 4), (, 509)} f () i not etermine a tetraheron, suggesting that the minimum value for x is between an an, since f() i not etermine a tetraheron, the maximum value for x is between an An, f (9) suggests a maximum volume of about 5 u when the istance is close to 9 u Subset of f similarly escribes the volume of the tetraheron as a function of the istances between V 0 an V Subset is a focus upon approximating the minimum istance as the volume approaches a minimum of 0 u S {(, 49), (, not), (, 590), (, not), (, 009), (, not), (7, 004), (, not), (9, 00), (, not), (7, 004), (, not),}, not inicating that the value of x i not prouce a tetraheron (iscounting egenerate tetraherons) Subset of f escribes the volume of the tetraheron as a function of the istances between V 0 an V Subset is a focus upon approximating the maximum istance as the volume approaches 0 u S {(, 07), (9, not), (, 95), (7, not), (4, 0), (5, not), (4, 09), (47, not), (4, 0), (4, not), (49, 004), (40, not)}

7 Graph epicts the approximate minimum x, the approximate maximum x, an the associate volumes as (V 0, V ) varies, f Graph Approximate minimum an maximum istances between the vertices V 0 an V an the aproximate maximum volume of the tetraheron etermine by (7,9,,0,,x) associate with f