New methods for
the three-dimensional representation

of
high-dimensional objects

Philip Van Loocke

** **

The question how high-dimensional structures can be visualized in three dimensions has been raised since the advent of non-Euclidean geometry and of modern physics (Kaku, 1994; Barrow, 2003), and has inspired several artists (Henderson, 1983). Toward the end of the nineteenth century, knowledge about the conceivability of a fourth dimension spread, among others, in England and in France. From about 1910, cubist art was often explained as a means for the artist of expressing a four-dimensional reality. Artists such as Duchamp explicitly tried to develop new methods for visualizing the fourth dimension. They embedded their practice in a more global philosophical view on a four-dimensional universe. It was influenced by Russian authors and artists (such as Ouspensky and Malevitch), who linked four-dimensional concepts with a mystical attitude, which was linked with platonism. In the United States, ideas about higher dimensions widely spread through literature, such as the science fiction of Wells. Through the work of Bragdon, four-dimensional concepts found their way to the design of ornaments. In the twenties, Buckminster Fuller brought these ideas into the field of architecture. But although there are notable exceptions, the four-dimensionalist artistic/philosophical inspiration and interest faded after 1920.

In the past few decades, this interest gradually re-emerged, because computer technology gave new opportunities for visualizing high-dimensional objects. Popular work on computer graphics, like the book of Banchoff (Banchoff, 1990), brought these possibilities under wider attention. Digital architecture included the concept of higher dimensions in its design process, such as in the work of Novak or Lalvani (Lalvani, 2003).

In this paper, new methods for obtaining
computer generated visualizations of high-dimensional objects are constructed.
In section 2, a stepwise algorithm for visualizations of n-dimensional
hypercubes is explored. The algorithm is run for metric constraints, and for
structural constraints that are cast in metric terms (section 2). Subsequently,
the algorithm is reformulated in terms of branching structures. This yields representations
in which different types of three-dimensional spatial structures are combined.
Section 3 comments on the difficulty of reconciling structure and metric in the
representation of high-dimensional objects. As an alternative to a stepwise
procedure, section 4 describes a structural approach that generates forms of a
more organic nature. Finally, section 5 describes how the visualizations of
hypercubes can serve as a basis for a transform of the entire space R^{n}
onto R^{3}. The transform is illustrated with visualizations of some
four-dimensional structures defined in quaternion space.

**2. A stepwise algorithm for two and three dimensional visualizations
of hypercubes**

Consider a hypercube in n dimensions. It has 2^{n}
edge points or ‘vertices’, which are denoted P_{i}. Suppose that the
hypercube is mapped on a two- or three-dimensional visualization. The point on
which P_{i} is mapped is denoted Q_{i}. The vertices P_{i}
are supposed to have coordinates equal to –1 or +1. The coordinates of Q_{i}
take continuous values and are adapted in the course of the stepwise algorithm.

Suppose that the coordinates of Q_{i} are initialized
randomly. The Euclidean distance d(Q_{i},Q_{j}) for all couples
of points (Q_{i},Q_{j}) is determined, and this quantity is
normalized by division by the largest value found for it: d*(Q_{i},Q_{j})=d(Q_{i},Q_{j})/d_{max},
where d_{max} is the maximal distance between points Q_{i} and
Q_{j}. Similarly, in the
hypercube, all n-dimensional Euclidean distances d(P_{i},P_{j})
between edge points are determined, and normalized by division by 2.n^{(1/2)}:
d*(P_{i},P_{j})=d(P_{i},P_{j})/ 2.n^{(1/2)}
(for the given choice of coordinates, 2.n^{(1/2)} is the largest
distance between different edge points in the hypercube). Then, the error
measure E_{1} = S_{{}_{i,j=1,…,n}_{}} (d*(Q_{i},Q_{j})-d*(P_{i},P_{j}))^{2} can be used to measure
the metric resemblance between the hypercube and its visualization. It is
proportional to a measure known as ‘stress’ or ‘Sammon error’ in the literature
on data visualization.

A stepwise algorithm minimizing this
error can be defined as follows. The points Q_{i} are give random
initialization. At each step, one point Q_{j} and one dimension in the
visualization are selected randomly. The quality e.a is added to the
coordinate of Q_{j} along this dimension; e is the stepsize of the
algorithm, and a is a random number between –1 and +1. If the new coordinates of Q_{j}
lead to a lower value of E_{1}, the change in coordinates is kept;
else, the old coordinate values are restored. Different runs of this algorithm
generally yield different visualizations, though the final value of E_{1}
is often the same. Figure 1 shows a two-dimensional visualization of a
three-dimensional cube (with E_{1}=1.904); Figure 2 illustrates a
two-dimensional visualization of a four-dimensional cube (with E_{1}=11.946).

**Figure 1**. Two-dimensional visualization
of a cube by minimization of E_{1}

**Figure 2.** Two-dimensional visualization
of a hypercube with n=4 and for minimization of E_{1}

The visualization of the hypercube in Figure 2 lacks symmetry.
Symmetry is restored when the algorithm is applied for three dimensional
visualizations. This is illustrated in Figure 3, where two representations
corresponding to n=4 are shown (in both cases, E_{1}=3.769). The
visualizations obtained show point-symmetry relative to the center of the Q_{i}-structure.
The left and the right part of Figure 4 show a representation of a five,
respectively a six-dimensional hypercube (with E_{1}=24.102 and E_{1}=120.378).
The point-symmetry of the visualization remains for n=5 and n=6.

**Figure 3**. Two visualizations of
hypercubes for n=4 and for minimization of E_{1}

**Figure 4**. Visualizations of hypercubes
for n=5 (left) and n=6 (right)

Consider the smallest squares on the surface of the hypercube. These
squares are determined by edge points {P_{i1},…,P_{i4}} for
which n–2 coordinates take the same value, and for which the remaining two
coordinates form the four combinations (–1, –1), (–1,+1), (+1, –1) and (–1,+1).
The number of such squares is equal to s(n) = 2^{n-2}. B(n, n–2), where
B denotes the binomial value. Similarly, the smallest 3-dimensional cubes are
formed by sets of different edge points {P_{i1},…,P_{i8}} which
have the same value along one dimension. The number of such cubes in the
hypercube is c(n) = 2^{n-3}. B(n, n – 3).

The four edge points of a smallest square define (4^{2 }–
4)/2 = 6 different distances. For the j-th smallest square, the k-th distance
between edge points is denoted ds_{jk}. Similarly, for every smallest
cube, the eight edge points define (8^{2 }– 8)/2 = 28 distances; dc_{jk}
is the k-th distance relation associated with the j-th cube. The largest
distance that occurs in a smallest square is 2^{(1/2)}; in case of a
cube, the corresponding value is 3^{(1/2)}. Similar distances can be
determined also for the images Q_{i} of the edge points P_{i}
in the visualization. Suppose that d’s_{jk} is the k-th distance
between the points in {Q_{j1},…,Q_{j4}} on which the square {P_{j1},…,P_{j4}}
is mapped. Similarly, d’c_{jk} is the k-th distance between points on
which the edge points of the j-th smallest cube are mapped. The largest distance in d’s_{jk}
(k=1,…,6), respectively d’c_{jk} (k=1,…,28), is denoted d’s_{j,max},
respectively d’c_{j,max}. With
these notations, two additional error functions can be defined:

E_{2}= S_{{j=1 to s(n)}} S_{{k=1 to 6}} (ds_{jk}/2^{(1/2)}_{ }– d’s_{jk} /
d’s_{j,max} )^{2}

E_{3}= S_{{j=1 to c(n)}} S_{{k=1 to 28}} (dc_{jk}/3^{(1/2)}_{ }– d’c_{jk} /
d’c_{j,max} )^{2}

Error function E_{2} measures how well smallest squares in a
hypercube are mapped on squares in the visualization. E_{3} measures how
well the visualization preserves distance relations for smallest cubes. Figure
5 shows two examples of two-dimensional solutions found for minimization of E_{2}
and for n=3 (and with E_{2}=12.618 and E_{2}=8.657). For n=3, E_{3}
coincides with E_{2}. Figure 6 shows a visualization of a hypercube
with n=4 and for minimization of E_{2} (E_{2}=22.952). In Figure 7, E_{3} is minimized (E_{3}=21.661).
Three-dimensional visualizations are shown in Figures
8-11. Figure 8 has two examples of visualizations for n=4 and minimization of E_{2}
(E_{2}=12.394 and E_{2}=12.200). Figure 9 shows two examples
for minimization of E_{3} (E_{3}=12.629 and E_{3}=
12.618). The left, respectively the right, form in Figure 10 gives a
visualization for n=5 and minimization of E_{2}, respectively E_{3}
(E_{2}=53.965 and E_{3}=80.735). Finally, in Figure 11, the
left, respectively the right part, illustrates the case n=6 for minimization of
E_{2}, respectively E_{3} (E_{2}=173.245 and E_{3}=365.124).

**Figure 5**. Two visualizations for cubes
obtained by minimization of E_{2}

**Figure 6**. Two-dimensional visualization
of a hypercube for minimization of E_{2}

**Figure 7**. Two-dimensional visualization
of a hypercube for minimization of E_{3}

**Figure 8**. Three-dimensional
visualization of the hypercube for n=4 and minimization of E_{2} .

**Figure 9**. Visualization of the hypercube
for n=4 and minimization of E_{3}

**Figure 10**. Visualization for n=5 and
minimization of E_{2} (left) and E_{3} (right)

**Figure 11**. Visualization for n=6 and
minimization of E_{2} (left) and E_{3} (right)

**3. Angular variables corresponding to a branching process**

** **

In the previous section, the search algorithm varied Cartesian
coordinates of edge points Q_{i}. For a different choice of variables,
the points Q_{i} can be obtained by a spatial branching process
consisting of n subsequent bifurcations. At the first stage of the bifurcation
process, two branches with starting point in the origin are defined. The
endpoint of a branch is the starting point for two new branches at the next
level, until, at stage n, 2^{n} endpoints are generated. Each branch is
characterized by two angles f and q, and by a length l, where f is the angle
between the horizontal projection of the branch and the x-axis, and q is the angle between
the branch and the z-axis. The angles f and q can be adapted by the stepwise algorithm of
the previous section, for any of the functions E_{1}, E_{2} or
E_{3}. This yields a system with 2. (2+…+ 2^{n}) angular
parameters. In the examples in Figures 12-13, branches were given length
inversely decreasing with the branching level.

The configuration to which the branching system converges lacks
symmetry, but the endpoints of the branches have point-symmetry. This accords
with the fact that the value of E_{1} in these examples approximates
the values found in section 2 (in Figure 12, E_{1}=3.801, and in Figure
13, E_{1}=24.503). The algorithm can be run for symmetrically
constrained branching systems. In this case, half of the tree is constructed as
a point-mirror of the other half, and the total number of parameters is divided
by 2. Then, solutions with higher E_{1}-values are obtained. This is
illustrated in Figures 14-15, where symmetry was imposed, resulting in E_{1}-values
10.757, 12.624 (Figure 14) and 65.002, 48.916 (Figure 15). The illustrations
are for branches of constant length (except for the first two branches, which
were given shorter length).

These visualizations of hypercubes have
a special aesthetic property. The tree-structure in itself has no clear
interpretation to an observer not informed about its purpose. In Figures 14-15,
the visualization of the hypercube, when drawn apart from the underlying
tree-structure, is also of limited appeal (since the E_{1}-value is
relatively high, the fact that a hypercube is being visualized is not
apparent). But by matching the tree-structure with the with the visualization
of the hypercube, each type of three-dimensional structure can be interpreted
with help of the other.

**Figure 12**. Two examples for angular
variables and with branches of decreasing length

**Figure 13**. Visualization of the
five-dimensional hypercube for angular variables

**Figure 14**. Visualizations of hypercubes
with point symmetry imposed

**Figure 15**. Visualizations of hypercubes
(n=5) with point symmetry imposed

The branching algorithm can be run in a sequential way. First, a
visualization for n=2 (corresponding to the visualization of a square) is
computed. The 2^{3} additional branches required for the visualization
of the cube (corresponding to n=3) are attached at the endpoints of the
branches corresponding to n=2. During the search corresponding to n=3, the
latter are kept fixed. This process is continued. At the k-th step, the angles
associated with the 2^{k} new branches are allowed to vary. This
results in a fast algorithm when compared to a non-sequential procedure. Such
an algorithm was run several times. Branching structures with systematicity
were never found, in the sense that angles between branches at later stages
were not related in any systematic way to angles between branches at previous
stages. This was not helped if different dimensions were given different weight
in the metric function. Suppose that the coordinates of P_{i},
respectively Q_{i}, are denoted x_{i}^{k }(k=1,…,n),
respectively y_{i}^{k} (k=1,2 or k=1,2,3). In Figure 16, the
error function E’_{1} = S_{{i,j=1 to N}} (d^{n}(P_{i},P_{j})/d_{max }– d^{3}(Q_{i},Q_{j})/d’_{max})^{2},
with d^{n}(P_{i},P_{j}) = (S_{{}_{k=1 to n}} ((1/k).(x_{i}^{k}–x_{j}^{k})) ^{2})
^{(1/2)} and d^{3}(Q_{i},Q_{j}) = (S_{{}_{k=1 to 3}} ((1/k).(y_{i}^{k}–y_{j}^{k}))^{2})^{(1/2)
}was used. As a consequence, each additional dimension led to a smaller
deformation of the metric relations obtained at a previous stage.

**Figure 16**. Tree-visualization for n=7
and with E’_{1 }= 126.390

The lack of systematicity limits the aesthetics of the branching forms. Structural properties appear to be hard to reconcile with metric constraints, and this difficulty increases as n increases. For values of n higher than 6, representations which are visually accessible, or which have aesthetic symmetry, can still be obtained if the metric procedure is replaced by structural considerations.

**4. Structural visualizations of high-dimensional discrete spaces**

The spatial branching process used in
section 3 can be generalized into an alternative visualization procedure based
on structural considerations. As a point of departure, a rectangular parallelepiped
is constructed with sides b.g, b.g^{2} and b.g^{3} (g<1). This structure is turned into a visualization of
four-dimensional hypercube by splitting the smallest side-surfaces into two
smaller rectangles with sides b.g^{4} and b.g^{5} (see Figure 17). Then, the new rectangles split again in two smaller
rectangles to yield a representation of a five-dimensional hypercube, and so
on. For splitting processes with appropriate structural properties, aesthetic
representations result.

**Figure 17**. Splitting the upper
side-surface of a rectangular parallelepiped

An instance of a splitting process is defined as follows. We confine the definition to the splitting process at the top rectangle of the cube (so that half of the visualization of the hypercube is generated; the other half is obtained by mirroring the resulting structure relative to the horizontal plane).

The first three binary dimensions of the
n-hypercube are associated with the edges of the parallelepiped. Then, two
curves are constructed, corresponding with fourth component values +1 and –1,
respectively. The curves are defined as successions of t line segments. The
starting point of the curves is the middle of the upper rectangle of the
parallelepiped. The angles j and q of the segments linearly increment according to the rule: dj=c.x_{4}
and dq=d.x_{4}, where x_{4 }is the fourth component value
associated with a curve. The new rectangles are attached on top of these
curves. They are obtained by scaling the top-surface of the parallelepiped, and
are orthogonal to the segment on top of which they are drawn. The length of a
line segment in a curve is multiplied with a factor z (z <1) at each step, so
that the bifurcating curve structure has smaller branches after each new
bifurcation.

After the rectangles corresponding to
the fourth dimension are drawn, each of the curves splits into two new
curves. The change in angles of the
segments in the curves remains determined by dq = c.x_{k} and dq = d.x_{k}
(k³5). This process is continued until k equals n. In order to allow
endpoints to be located in different horizontal planes, at each odd step in the
bifurcation process, the value of q is increased by an amount k at the initial
segment corresponding to x_{k }= 1 if x_{4 }= 1, and at the
initial segment corresponding to x_{k }= -1 if x_{4 }= -1. In
order to prevent the branching structure from curling too strongly onto itself,
the parameter k is defined as a decreasing function of k. More specific, in the
illustrations in Figures 18-19, k=k_{c}(n+1-k)^{2}. Figure 18 shows the upper part of the visualization
of a 13-dimensional hypercube for c=0.0175, d=0.0175, z=0.993, and k_{c}=0.125. Figure 19 shows the underlying curve structure for variation
of one parameter in the construction (d=0.0105). Figure 19 was drawn with
continuous circular contours in order to ease visual track of the bifurcation
process.

**Figure 18**. Upper part of a structural
visualization of the thirteen dimensional hypercube

**Figure 19**. Example of an underlying
bifurcating curve set

The aesthetics of bifurcating curve sets can be studied in its own
right. For different parameters, and for a constant value of k, a binary tree
corresponding to ten bifurcations is shown in Figure 20. The end-points of this
structure are associated with a binary, ten-dimensional space. The procedure
can be defined for p-ary spaces. The nature of the curve sets generated can be
made more general also by giving the rules for dj and dq a more general
form. This is illustrated for a ternary space, and for which x_{k} can
take values –1, 0 and +1 in Figures 21-25 (k=1,…n; since we only illustrate the
branching curve structure, k can start at value 1 instead of at value 4).
Suppose that dj and dq are defined in accordance with dj = c.f(j,q).x_{k}
and dq = d.g(j,q).x_{k} (c and d are
constants; also k is kept constant in the illustrations which follow). If f and g are
well chosen, end-points of branches develop in non-co-planar ways and with
aesthetic spatial symmetry. The parameters and functions in these rules can be
tuned toward a visually accessible representation of a p-ary space, or toward
representations of aesthetic interest.

The definition of f and g is facilitated if the angles j and q are drawn into
the interval [0,p/2]. Two angles µ and n are considered; µ is obtained from j by subtracting
iteratively p/2 from Abs(j) until a number in [0,p/2] is obtained; n is obtained in the same
way on basis of q. Then, the form in Figure 21 is obtained for f=-µ and g=n (Figures 21-24 show branching structures with endpoints corresponding
to the points of a seven-dimensional ternary space). The form in Figure 22
results with f=µ^{2} and g=1-n. Figure 23** **is obtained for** **f=8µ^{(1/2)} and g=-n^{(1/2)}** **. Figure 24 corresponds f=µ^{4}
and g=µ^{4}. Figure 25 is made for a ternary eight-dimensional space
and for f(m)=-m^{1/2},
and g(n)=8n^{1/2}.

**Figure 20**. Form associated with a space of ten binary
dimensions

**Figure 21**. Form with f=-µ and g=n

** Figure 22**. Form with f=µ^{2} and g=1-n

** Figure 23.** Form with f=8µ^{(1/2)} and g=-n^{(1/2)}

**Figure 24**. Form with f=µ^{4} and
g=µ^{4}

**Figure 25**. Form corresponding to an
eight-dimensional ternary space

In Figures 20-25, the curve-structure was enveloped with non-circular volumetric elements. For each curve, circles were drawn around the end-points of all line segments, and orthogonal to the curve. In case of a ternary space, for each line segment, there are two corresponding segments on alternative curve-continuations and which belong to the same stage in the bifurcation process. An enveloping circle was elongated in the direction of these alternative continuations. The resulting contours were connected by lines.

** **

**5. Transforms constructed as weighted vertex-based functions **

** **

The previous section aimed to obtain
visualizations for high-dimensional spaces by exploration of the structural
instead of the metric domain. This section returns to the metric visualizations
of section 2. Instead of visualizing hypercubes, a general transform F: R^{n} ® R^{3} is described which visualizes arbitrary
high-dimensional objects. The transform takes a

visualization of the hypercube as its
point of departure, and maps a point in R^{n} on a point in R^{3}
as a function of its metric and angular relations with the vertices (or ‘edge
points’) and the edges (or ‘side-lines’) of the hypercube. The visualizations
obtained by the algorithm of section 2 generally do not correspond to linear
projections of the hypercube. Therefore, the function F which embeds objects in the
visualization is also not linear, although usually only ‘weakly’ so.

The transform F is defined as a weighted
sum of angular vertex-based transforms. With each vertex of a hypercube, an
angular vertex-based transform f_{k}(x) is associated as follows (k=1,…,2^{n}).
The unit vector originating in the center of the hypercube and pointing to the
k-th edge point is denoted e_{k} (k=1,…,N). The components of e_{k} are equal to 1/n^{(1/2)}
or –1/n^{(1/2)}. The unit vector pointing from Q_{k} to Q_{r}
in the visualization is denoted v_{kr}. With these notations, f_{k}:
R^{n} ® R^{3} associated with vertex P_{k}
is defined as follows:

f_{k}(x) = Q_{k }+ s. |x–P_{k}| . (S_{{}_{r}_{=1,...,N; }_{r}_{¹}_{k}_{} }(e_{r}. (x–P_{k})/ |x–P_{k}|
) . v_{kr}) / c

with:

c= | S_{{}_{r}_{=1,...,N;
r}_{¹k}_{} }(e_{r}. (x–P_{k})/ |x–P_{k}| ) . v_{kr }|

The function f_{k}(x) contains a distance factor |x-P_{k}|
and a directional factor (S_{{}_{r}_{=1,...,N}
}(e_{r}. (x–P_{k})/
|x–P_{k}| ) . v_{kr}) /
c. Due to the normalization, the directional factor specifies a unit vector in
R^{3}. The distance factor entails that the distance between f_{k}
(x) and Q_{k} in R^{3}
is s times the distance between
x and P_{k} in R^{n}, from which it follows that f_{k}(P_{k})=Q_{k}.

Figure 26 illustrates an edge-based transform for n=4 and for the
point Q_{1} in the Q_{k}-structure in the first form in Figure
8. The Figure shows the transform of a 12x12 grid defined over the
four-dimensional hypercube. All points on the grid were transformed according
to f_{1}(x) (with s=1.05. a/b, where a is the average distance between edges in the P_{k}-hypercube,
and b is the average distance between different points Q_{k}). It
can be observed that, in the neighborhood of Q_{1}, the transformed
grid approximates a linear grid structure coinciding with planes through Q_{1}
and the neighbors with which it forms a smallest square. The quality of the
approximation decreases for points which are images of points removed from P_{1}
in the hypercube.

**Figure 26. **Vertex-based transform
operating on the side-planes of a four-dimensional hypercube containing Q_{1}.
The left figure shows the transforms of the side-squares containing Q_{1}.
The right figure shows the transform of the entire side-square structure.

A good approximation of an overall linear structure over the Q_{i}-structure
is obtained if vertex based transforms for all points P_{i} are summed
into a weighted combination. The function F is obtained if a transform
f_{k} is given a weight proportional to a power r of the inverse
distance between x and P_{k}:

F(x) = (1/c). S_{{k=1 to N} }f_{k}(x) / |x-P_{k}|^{r}

with

c= S_{{k=1 to N} }1 / |x-P_{k}|^{r}

Figure 27 shows the result of F(x) when applied to the
same Q_{k}-structure as in Figure 26. It can be noticed that the voted
sum turns local approximations into global approximations (the Figure was
generated for r=1; this property remains for other visualizations of the hypercube,
and for values of n higher than 4; a study of this fact was carried out in Van
Loocke, 2003).

**Figure 27**. F(x) operating on the
side-squares of the hypercube

The function F(x) can be used to obtain visualizations of any n-dimensional form. In Figures 28-33, it was applied on a generalization of a function familiar in a fractal context.

Suppose that a four-dimensional point x
has coordinates x_{1},…,x_{4}. Then, the quaternion expression
f(x)=x^{2}+c has four components f_{1},…,f_{4} which
are defined by f_{1}(x)=x_{1}^{2}-x_{2}^{2}-x_{3}^{2}-x_{4}^{2}+c_{1}, f_{2}(x)=2x_{1}^{2}x_{2}^{2}+c_{2},
f_{3}(x)=2x_{1}^{2}x_{3}^{2}+c_{3},
f_{4}(x)=2x_{1}^{2}x_{4}^{2}+c_{4}.
The following procedure defines a function g(x). First, f(x) is
iterated t_{1} times. The result of this function, but with the first
component transformed in its negative, is used as input for t_{2}
additional iterations of f, after which the first component is changed into its
negative again. The resulting point is uniformly contracted toward the origin,
except for the first coordinate, which is displaced by an amount a. This cycle of
t_{1}+t_{2} iterations is repeated t_{3} times. The
resulting point is g(t_{1},t_{2},t_{3};x). In Figures 28-33, gg(t_{1},t_{2},t_{3};x)
was applied to a square 27x27-grid defined over a structure defined over the
four-dimensional hypercube which was contracted uniformly to the origin by a
factor h. In Figure 28, h =0.333, t_{1}=2, t_{2}=1 and t_{3}=3. In
Figure 29, the same parameters are used, but F is applied on basis of a
different visualization of the hypercube. In the next Figures, these parameters
were h =0.333, t_{1}=3, t_{2}=2 and t_{3}=3
(Figure 30); h =0.333, t_{1}=3, t_{2}=4 and t_{3}=3.(Figure
31); h =0.333, t_{1}=5,t_{2}=4 and t_{3}=4 (Figure
32); h =0.0121, t_{1}=5, t_{2}=5 and t_{3}=6
(Figure 33); h =0.0121, t_{1}=5, t_{2}=5 and t_{3}=8
(Figure 34).

**Figure 28 **(left)**
and Figure 29 **(right)

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**Figure 30** (left) and **Figure 31** (right)

**Figure 32 **(left)** and Figure 33 **(right)

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**Figure 34 **(left)** and Figure 35 **(right)

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**Figure 34 **(left)** and Figure 35 **(right)

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**Figure 36 **(left)** and Figure 37 **(right)

** **

Variations of F(x) can be obtained in several ways. In Figures 35-36, F(x) was applied
on the side grid over the hypercube, but for functions f_{k}(x) in
which the factor s|x-P_{k}| was replaced with s|x-P_{k}|.(1+sin(5|x-P_{k}|)).
In Figure 37, this strongly non-linear transform was used to map the side grid
after it had been transformed by the procedure used in 28-34, and for h=0.4, t_{1}=1,
t_{2}=1 and t_{3}=4.

** **

**6. Conclusion**

The paper considered visualizations of high-dimensional objects. As a point of departure, visualizations were used which came out of stepwise algorithms. The tension between metric accuracy and desirable structural properties was discussed. In section 5, a non-linear projection method was developed to obtain visualizations of high-dimensional objects matched in the visualization of a hypercube. In other work, the latter method was applied to data-visualization and was compared with principal component analysis, which corresponds to the selection of a well-chosen linear projection. It was pointed out that, for different real-world data-sets, the metric accuracy of a visualization based on the non-linear transformation F was better than the metric accuracy of principal component analysis (Van Loocke, 2003). This is part of the motivation for using F instead of linear projections in section 5.

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