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The Chameleon transformation in n
n
n
dimensions
Dimitris Vartziotis∗†‡
, Florian Schneider∗
, Diana Behnke∗
, Grace Tian ∗§
September 12, 2023
Abstract
We present the idea of ”Chameleon” transformations, providing a mathematical
framework to study the interaction between an individuum and its environment.
Through iterations of the transformation and its variants in two, three, and ar-
bitrary spatial dimensions, we explore the dynamics of this interaction between
individuum and environment, while focusing on the influence of the environment on
an individuum. For a simplex surrounded by a sphere, we establish convergence.
We find a transfer of regularity from the environment to the individuum. Numerical
results support this finding, suggesting linear convergence rate. Future research di-
rections include investigating the impact of anisotropic environments on regularity,
extending the framework to meshes of triangles, and exploring the reverse process
of the individuum affecting the environment. Insights into the interplay between
the individuum and its environment will enhance our understanding of mathemat-
ical modeling in real-world phenomena. Chameleon transformations offer potential
applications in various fields, including mesh smoothing.
Keywords: linear algebra, geometry, triangle transformations, polygon transformations,
mesh regularization.
1 Introduction
Mathematics is often considered to be a universal language that allows us to model and
understand the world around us. It has long been recognized as a powerful tool for describ-
ing physical phenomena, from the motion of celestial bodies to the behavior of subatomic
particles. However, the question remains: does mathematics have inherent properties that
reflect the world we observe? Is there a deeper connection between mathematics and the
natural world, or are these two realms entirely separate?
∗TWT GmbH Science & Innovation, Stuttgart, Germany
†NIKI Ltd. Digital Engineering, Ioannina, Greece
‡Corresponding author, E-mail address: dimitris.vartziotis@nikitec.gr
§Massachusetts Institute of Technology, Cambridge, MA, USA
1
In this paper, we propose a new perspective on the role of mathematics in understanding
the world. We argue that every individual is part of a larger environment, and that the
sociocultural context in which we live affects the way we perceive and interact with the
world. This adaptation of individual to environment is a key occurrence in nature, at
the organism level [32], the species level [33], and beyond. Similarly, the impact of an
individual on its environment is seen universally from the micro [1] to the macro [2] scale.
In the sociocultural context, we see individuals adapting to their environment through
social conformity [20] and cultural assimilation [31]. Similarly, individuals can shape their
environment both socially [21] and culturally [9].
We develop the idea of an individual-environment interaction in the context of mathemat-
ics by introducing the Chameleon transformation1on arbitrary triangles. Specifically, we
consider the interaction of a triangle, representing the individuum, with a circle, repre-
senting the environment. Through this interaction, we are able to regularize the triangle,
reflecting the ways in which an individual can be influenced by its environment.
We prove the convergence of this transformation in arbitrary dimensions, demonstrat-
ing the robustness and universality of the method. Additionally, we discuss numerical
experiments that illustrate the effectiveness of this approach in various settings.
By using the triangle-circle interaction as a concrete example, we aim to highlight the
broader implications of the individual-environment dynamic in mathematics. We suggest
that this dynamic is not unique to triangles and circles, but is a fundamental aspect of
mathematical modeling and understanding more generally.
Furthermore, we explore the idea that a ”simple” reversal of the mathematical model
shows that the individual can also affect the environment in significant ways. This obser-
vation serves as a motivation for everyone to work at their best, to use mathematics to
understand and improve the world around them.
Overall, we hope to show that mathematics is not simply a tool for describing the world,
but a complex and dynamic system that reflects and shapes our understanding of the nat-
ural world and our place within it. By exploring the sociocultural context of mathematics
and its relationship to the individual and the environment, we hope to gain new insights
into the nature of mathematical thought and its role in shaping our understanding of the
world.
The paper is organized as follows: Section 2 introduces the motivation for the method
and provides two examples of Chameleon transformations. Section 3 contains a descrip-
tion of the generalized Chameleon transformation and proofs of its nondegeneracy and
convergence. Section 4 provides a numerical analysis for the convergence rate of the iter-
ation. Section 5 presents conclusions and outlook for future research. Finally, A contains
additional proofs and B contains further numerical results.
1The idea of Chameleon transformation was originally proposed by Dimitris Vartziotis.
2
2 The chameleon transformation in two dimensions
2.1 Motivation
Consider the triangle ∆ := conv ({vA,vB,vC})∈R2.1For simplicity, assume
vA,vB,vC∈ S1:= {x∈R2:∥x∥2= 1}(otherwise take the circumcircle of the tri-
angle, move it to the origin, and scale accordingly).
Definition 2.1. Let O ⊂ R2be the one-dimensional boundary of a closed and bounded
domain Ω⊂R2, such that it completely surrounds the triangle, i.e.,
∆⊊int (Ω) .(2.1)
We call Othe environment.
We are interested in the interaction of the triangle with the environment. Specifically,
we investigate how the environment shapes the triangle. Thus, we look at the general
iteration
∆(k+1) =TO∆k,
where TOmaps a triangle to another triangle. In the following, we focus on the case where
O=R·S1:= {x∈R2:∥x∥2=R}, where 1 < R ∈R. As such, we will drop the index
of the transformation and use the shorthand, T=TO.
Example 2.2. Let ¯vA∈ O be the first intersection of the half ray from vAin the direction
of vB−vA, i.e., there exists a minimal ε > 0such that ¯vA=vA+ε(vA−vB). The
existence of such an εfollows directly from (2.1). Define ¯vB,¯vC∈ O similarly (permuting
the vertices accordingly).
We then project ¯vi, for i∈ {A, B, C}, back onto S1, i.e. we re-define
˜vi:= ¯vi
∥¯vi∥2
.(2.2)
The iteration is then defined to be
T(∆) = TC◦TB◦TA(∆) .(2.3)
One sub-step of this procedure (for one vertex) is shown in Figure 2.1 (a).
Example 2.3. For vi,i∈ {A, B, C}, define pi∈R2as the orthogonal projection of
vionto the affine subspace Sispanned by the other two vertices, i.e. vj∈Si, j ∈
{A, B, C }, j =i.
Let ¯vi∈ O be the first intersection of the half ray from viin the direction of vi−piwith
the environment, i.e., there exists a minimal ε > 0such that ¯vi=vi+ε(vi−pi). The
existence of such an εfollows directly from (2.1).
1Here and below we consider Rnas a vector space and don’t make a difference between a point and a
vector that is connecting the origin and this point.
3
vA
vB
vC
¯vA
˜vA
O
(a)
vA
vB
vC
¯vA
¯vB
¯vC
˜vA
˜vB
˜vC
O
(b)
Figure 2.1: (a) Sub-iteration TAof Chameleon transformation (2.3). (b) One full iteration
T∗of simultaneous transformation (2.6).
We then project ¯vi, for i∈ {A, B, C}, back onto S1, i.e. we re-define
˜vi:= ¯vi
∥¯vi∥2
.(2.4)
The iteration is then defined to be
T(∆) = TC◦TB◦TA(∆) .(2.5)
One sub-step of this procedure (for one vertex) is shown in Figure 2.2.
vA
vB
vC
¯vA
˜vA
pA
O
.
Figure 2.2: Sub-iteration TAof Chameleon transformation (2.4).
Remark 2.4. Both (2.2) and (2.4) can be used consecutively per vertex (as in (2.3) and
(2.5)) or simultaneously, in the sense that
T∗(∆) = conv ({˜vA,˜vB,˜vC}).(2.6)
See Figure 2.1(b) for an illustration of that in the case of (2.2). Numerically, we observed
similar convergence behaviour for both consecutive and simultaneous iterations and thus
focus on the consecutive version due to its structural simplicity.
4
Any triangle under the iterations (2.3) or (2.5) converges to a regular triangle when
the environment is regular (see proofs in Theorem 3.7 and Theorem A.1, and numerical
results in Section 4). In other words, the transformations force the triangle to mimic
the properties of its environment. We therefore call such transformations Chameleon
transformations.
3 The chameleon transformation in arbitrary dimen-
sions
Let d≥2 and vi∈ Sd−1⊂Rd,i= 0, . . . , d. We define the d−Simplex ∆ :=
conv ({vi:i= 0, . . . , d}).
We define the regularizing iteration by
˜vi:= vi+ε(vi−pi)
∥vi+ε(vi−pi)∥2
(3.1)
where 0 < ε ∈Rand pi∈Rddenotes the projection of vionto the affine subspace Si
spanned by the other vertices, i.e. vj∈Si, j = 0, . . . , d;j=i.
We define the iteration deforming the simplex at vertex ias
Ti(∆) := conv ({vj:j= 0, . . . , d, j =i}∪{˜vi}) (3.2)
and the full iteration as
T(∆) = Td◦Td−1◦ · ·· ◦ T1◦T0(∆) .(3.3)
Remark 3.1. If we choose εas in Example 2.3, (3.3) becomes the d-dimensional equiva-
lent of (2.5) and is thus a Chameleon transformation.
Lemma 3.2. The iteration (3.1) is well-defined, i.e. vi+ε(vi−pi)=0.
Proof. Since piis the orthogonal projection of vionto Si, the vector x−piis orthogonal
to (1 + ε)(pi−vi) for each point x∈Si. By Pythagoras,
∥x−vi−ε(vi−pi)∥2
2=∥(x−pi) + (1 + ε)(pi−vi)∥2
2=∥(x−pi)∥2
2+∥(1 + ε)(pi−vi)∥2
2
for each point x∈Si. Suppose now vi+ε(vi−pi) = 0. This implies (1+ε)(pi−vi) = 1+ε
εvi.
Therefore,
∥x∥2
2=∥x−pi∥2
2+(1 + ε)2
ε2∥vi∥2
2∀x∈Si.
Since ∥vi∥2= 1, it follows that ∥x∥2≥1+ε
ε∥vi∥2>1 for each point x∈Si. On the other
hand, we can take vj∈Sifor j=iwith ∥vj∥2= 1. This is a contradiction.
Lemma 3.3. It holds that
⟨˜vi,vi−pi⟩≥⟨vi,vi−pi⟩(3.4)
with equality if and only if pi=αvi,α∈R.
5
Proof. Assume that pi=αvifor some α∈R. Then it holds that
˜vi
(3.1)
=vi+ε(vi−αvi)
∥vi+ε(vi−αvi)∥2
=1 + ε(1 −α)
∥vi+ε(vi−αvi)∥2
vi=1 + ε(1 −α)
|1 + ε(1 −α)|
vi
∥vi∥2
=
= sign (1 + ε(1 −α)) vi.
Assume that α≤1⇒1 + ε(1 −α)>0, implying vi=˜viand consequently
⟨˜vi,vi−pi⟩=⟨vi,vi−pi⟩. On the other hand, if α > 1, the hyperplane Si, which
is orthogonal to viand passes through pi=αvihas no intersection with Sd, which is a
contradiction since vj∈Si∩ Sdfor all j=i.
For the following, assume that pi=αvi, i.e., viand vi−piare linearly independent.
Define
z:= ⟨vi,vi−pi⟩
∥vi−pi∥2
,
w:= z+ε∥vi−pi∥2.
From Cauchy-Schwarz inequality, it follows that
⟨vi,vi−pi⟩2CSI
<∥vi∥2
2· ∥vi−pi∥2
2
vi∈Sd
=∥vi−pi∥2
2,
where the inequality is strict since viand vi−piare linearly independent. Thus, −1<
z < 1 and z < w.
Using the binomial theorem, it follows straightforwardly that
∥vi+ε(vi−pi)∥2
2=∥vi∥2
2+ 2ε⟨vi,vi−pi⟩+ε2∥vi−pi∥2
2=
= 1 + 2εz ∥vi−pi∥2+ε2∥vi−pi∥2
21−z2+w2.(3.5)
Additionally,
w· ∥vi−pi∥2=z· ∥vi−pi∥2+ε∥vi−pi∥2
2=⟨vi,vi−pi⟩+ε∥vi−pi∥2
2
=⟨vi+ε(vi−pi),vi−pi⟩(3.6)
Consequently,
⟨vi,vi−pi⟩=z∥vi−pi∥2=z
√1−z2+z2∥vi−pi∥2
(∗)
<w
√1−z2+w2∥vi−pi∥2=
(3.5)
=w
∥vi+ε(vi−pi)∥2∥vi−pi∥2
(3.6)
=⟨vi+ε(vi−pi),vi−pi⟩
∥vi+ε(vi−pi)∥2
(3.1)
=⟨˜vi,vi−pi⟩,
where we used in (∗) that the function w7→ w
√1−z2+w2is strictly increasing for in wfor all
|z|<1 and that z < w.
Lemma 3.4. The i-th substep (3.2) of the transformation T(∆) in (3.3) satisfies
vold(Ti(∆)) ≥vold(∆) with equality iff ˜vi=vi. In particular, vold(Ti(∆)) ≥vold(∆)
with equality iff ˜vi=vifor all i= 0, . . . , d.
Additionally, if vold(Ti(∆)) = vold(∆) then ˜vi=viand there exists such α∈Rthat
vi=αpiand ⟨vi,vj⟩=α∥vi∥2
2=αfor all j= 0, . . . , d with j=i.
6
Proof. The d-dimensional volume of ∆ can be written as
vold(∆) = 1
dvold−1(∆i) dist(vi, Si).(3.7)
Moreover, since vi−pi
∥vi−pi∥2is a unit normal vector to Si, we can see that
dist(vi, Si) = |⟨vi−pi,vi−pi⟩ |
∥vi−pi∥2
=⟨vi−pi,vi−pi⟩
∥vi−pi∥2
(3.8)
dist(˜vi, Si) = |⟨˜vi−pi,vi−pi⟩|
∥vi−pi∥2≥⟨˜vi−pi,vi−pi⟩
∥vi−pi∥2
,(3.9)
where in the first equality the absolute value can be dropped since pibeing the orthogonal
projection of vion Siimplies dist(vi, Si) = ∥vi−pi∥2.
It then follows from (3.4) that
⟨˜vi−pi,vi−pi⟩=⟨˜vi,vi−pi⟩−⟨pi,vi−pi⟩(3.4)
≥ ⟨vi,vi−pi⟩−⟨pi,vi−pi⟩=
=⟨vi−pi,vi−pi⟩,(3.10)
from which we can conclude
dist(˜vi, Si)(3.9)
≥⟨˜vi−pi,vi−pi⟩
∥vi−pi∥2
(3.10)
≥⟨vi−pi,vi−pi⟩
∥vi−pi∥2
(3.8)
= dist(vi, Si).(3.11)
The base of the simplex is not transformed after an iteration Ti(∆), because the vertices
belonging to Siare not changed. This implies that the volume of the base vold−1(∆i)
in (3.7) also remains the same. From this and (3.11) it is straigthforward to see that
vold(Ti(∆)) ≥vold(∆), with equality only if ˜vi=vi. The second part of the claim then
follows directly from Lemma 3.3.
Lemma 3.5. If ⟨vi,vj⟩=c∈Rfor all i, j = 0, . . . , d with j=i, then the simplex ∆is
regular.
Proof. It holds that
⟨vi,vj⟩=∥vi∥2∥vj∥2cos ∠(vi,vj)vi,vj∈Sd
= cos ∠(vi,vj).
If ⟨vi,vj⟩=c∈R, all enclosing angles are the same, from which the regularity of ∆
follows.
Corollary 3.6. For the sequence ∆(k+1) := T∆k,k= 0,...,∞, the sequence
(vold∆k)k∈Nconverges.
Proof. Applying Lemma 3.4 d+ 1 times yields
vold∆k≥vold∆k−1≥. . . ≥vold∆0,(3.12)
where the inequality is strict as long as ∆k= ∆k−1. Thus, the sequence (3.12) is mono-
tonically non-decreasing and bounded from above (since ∆k⊂ Sd) and therefore con-
verges.
7
Proposition 3.7. The sequence ∆(k+1) := T∆k,k= 0,...,∞converges and the limit
simplex is regular.
Remark 3.8. From Corollary 3.6, it follows from Lemma 3.4, that any limit simplex (if
existent) remains unchanged by the application of Ti. From Lemma 3.5 it is also regular.
For the existence of the limit simplex, it has to be ruled out, that the sequence ∆(k)for
large kdoes not become arbitrarily close to a regular simplex while rotating on the unit
sphere. While Lemma 3.4 guarantees, that every regular simplex is a fixed point of the
iteration Ti, it does not guarantee that the fixed point is actually attained.
We want to note, that numerical simulations have shown, that there appears to be a
limit simplex ˆ
∆, depending only on the initial positions of the vertices and their ordering.
However, a formal proof for this is still missing.
4 Numerical results
The convergence proof in Theorem 3.7 currently lacks an estimate on the convergence
speed of the iteration (3.3). In particular, the dependence on the environmental radius R
and the dimensionality of the problem dis of crucial importance. In the following section
and in the B, we investigate this for a variety of radii and d∈ {2,3}(i.e., triangles and
tetrahedra).
4.1 Experiment setup
Here, we focus on d= 2. For more information about d= 3, check B.
Initially, a triangle ∆0∈ S1, is created, see Figure 4.1(a). The initial coordinates of the
vertices are generated randomly in S1, where vA= (0,1) is fixed for simplicity. For the
remaining vertices, we choose uniformly random angles ϕB∈[0,π
3] and ϕC∈[π
3, π], in
order to get a position of a new point in polar coordinates, which are then transformed
to cartesian ones.
For this series of experiments the maximal number of iterations was chosen kmax = 30 and
the sequence of enviromental radii Riwas generated so that the Rmin = 1.1, Rmax = 10
and the Ri+1 −Ri= 0.1. We investigate the following sequence
tk= 1 −vold∆k
d
voldˆ
∆∈[0,1] (4.1)
in order to study the tendency of the volume of a randomly generated initial triangle to
the volume of a equilateral triangle. Here, tk= 1 implies a degenerate triangle while
tk= 0 corresponds to an equilateral triangle.
The values of tkprovided in this article were averaged over 100 realizations for each
environmental radius Ri.
8
(a) (b)
Figure 4.1: Example of the Chameleon iteration (2.4) for d= 2, O=R· S1,R= 1.3.
(a) An irregular triangle inscribed in the unit circle, vol2(∆0)/vol2ˆ
∆= 0.19. (b) The
transformed triangle after one iteration of (3.3), vol2∆k/vol2ˆ
∆= 0.84.
4.2 Experimental results
4.2.1 Convergence results
The numerical experiments have shown that the speed with which the vold∆kap-
proaches the volume of a regular triangle voldˆ
∆is directly proportional to the envi-
ronmental radius R. One can see this in Table 1 for d= 2 and Table 2 for d= 3, where
the initial value t0and first 14 iterations and selection of radii Rare listed.
9
HHHHH
H
k
R1.1 1.5 2 3 5 8 10
tkqktkqktkqktkqktkqktkqktkqk
0 7.6 ·10−10.26 7.7 ·10−10.62 7.8 ·10−10.73 7.8 ·10−10.84 8.0 ·10−11.09 7.9 ·10−11.77 7.9 ·10−12.15
1 3.3 ·10−10.85 1.2 ·10−10.97 5.4 ·10−21.00 2.2 ·10−21.03 1.1 ·10−21.15 1.0 ·10−20.74 1.1 ·10−20.55
2 2.3 ·10−10.92 3.4 ·10−20.99 7.2 ·10−31.00 1.0 ·10−31.03 1.2 ·10−41.13 5.0 ·10−61.00 1.2 ·10−61.93
3 1.7 ·10−10.95 1.0 ·10−21.00 9.5 ·10−41.00 4.4 ·10−51.03 6.3 ·10−70.78 1.8 ·10−81.56 1.1 ·10−80.52
4 1.2 ·10−10.97 3.0 ·10−31.00 1.3 ·10−41.00 1.7 ·10−61.03 1.8 ·10−90.83 6.5 ·10−11 - 1.2·10−12 -
5 9.1 ·10−20.98 9.0 ·10−41.00 1.6 ·10−51.00 6.1 ·10−81.02 1.9 ·10−11 1.19 1.1 ·10−14 - 1.2 ·10−14 -
6 6.8 ·10−20.99 2.7 ·10−41.00 2.1 ·10−61.00 2.0 ·10−91.02 3.8 ·10−13 - 1.5 ·10−16 - 8.3 ·10−17 -
7 5.1 ·10−20.99 8.0 ·10−51.00 2.7 ·10−71.00 6.0 ·10−11 1.00 3.8 ·10−15 - 4.1 ·10−17 - 4.0 ·10−17 -
8 3.8 ·10−20.99 2.4 ·10−51.00 3.5 ·10−81.00 1.7 ·10−12 0.93 5.7 ·10−17 - 3.7 ·10−17 - 8.1 ·10−17 -
9 2.8 ·10−21.00 7.2 ·10−61.00 4.4 ·10−91.00 4.8 ·10−14 - 7.2 ·10−17 - 1.4 ·10−17 - 8.5 ·10−17 -
10 2.1 ·10−21.00 2.2 ·10−61.00 5.8 ·10−10 1.00 1.6 ·10−15 - 7.4 ·10−17 - 4.3 ·10−17 - 6.2 ·10−17 -
11 1.6 ·10−21.00 6.5 ·10−71.00 7.2 ·10−11 1.00 3.2 ·10−17 - 2.8 ·10−17 - 4.5 ·10−17 - 8.2 ·10−17 -
12 1.2 ·10−21.00 2.0 ·10−71.00 8.9 ·10−12 1.00 7.9 ·10−17 - 4.0 ·10−17 - 9.7 ·10−17 - 1.4 ·10−16 -
13 9.1 ·10−31.00 5.8 ·10−81.00 1.1 ·10−12 1.01 4.1 ·10−17 - 4.3 ·10−17 - 1.1 ·10−16 - 1.7 ·10−16 -
14 6.8 ·10−31.00 1.7 ·10−81.00 1.3 ·10−13 0.97 3.7 ·10−17 - 3.1 ·10−17 - 1.7 ·10−16 - 1.6 ·10−16 -
Table 1: The values of tk(4.1) and convergence rate qk(4.2) with respect to iteration kand environmental radius R,d= 2.
10
The results, represented in the Table 1 show that the values of tkare decaying exponen-
tially and the speed of the decay is non-linearly depending on the environmental radius R.
For the better understanding of this phenomenon, one may refer to the Figure 4.2, where
the log10(vol2∆k+1−vol2∆k) for each iteration kand each Rare demonstrated.
Figure 4.2: log10 (vol2∆k+1−vol2∆k) with respect to the iteration kand environ-
mental radius R.
Comparing Figure 4.2 and Figure B.2 one can mention that the results are very similar
for the d∈ {2,3}. The number of iterations needed for the simplex to reach regularity,
and, consequently, for the method to converge, is decaying exponentially. In addition, it
is important to mention that for d= 3 case the decay of the values is less steep than for
the d= 2. An explicit calculation or estimation of the convergence rate for arbitrary d
remains an open question.
4.2.2 Convergence rate estimation
The convergence rate qwas estimated numerically using the Formula 4.2 below [13]:
q≈
log
vold(∆k+1)−vold(∆k)
vold(∆k)−vold(∆k−1)
log
vold(∆k)−vold(∆k−1)
vold(∆k−1)−vold(∆k−2)
(4.2)
The results of the convergence rate estimation can be seen in the Figure 4.3. For this
experiment the maximal environment radius was increased to Rmax = 25 in order to
investigate the phenomena more in detail. As in the Figure 4.2 above, after a certain
number of iterations for each environmental radius, it was not possible to estimate the
rate of convergence because the step size was reaching machine precision. The numerical
values of qare also represened in Table 1 and Table 2 so that one may compare the values
of tkand the corresponding values of qkfor each iteration k. One may observe that for the
most values of Rthe value of q≈1, meaning linear convergence. In the neighbourhood
of R= 10 the convergence rate estimations become unstable, demonstrating superlinear
convergence and then further stabilize at the region corresponding to R > 12. The same
phenomena is not observed in the Figure B.3, though. In the both cases the decrease
11
of the number of iterations needed for the step size to reach machine precision stabilizes
after a certain Rvalue, R= 7 for d= 2, and R= 9 for d= 3.
Figure 4.3: Rate of convergence qwith respect to the iteration kand environmental radius
R.
12
O
(a)
O
(b)
Figure 5.1: Simultaneous projections method with initial triangle ∆0drawn in black and
two separate elliptical environments of different dimensions. (a) The elliptical environment
Ohas major axis 8 and minor axis 6. After two iterations, we arrive at the red triangle,
with length ratio 0.89. The length ratio remains 0.89 (to the nearest hundredth) for 5
more iterations, at which point we consider it converged. (b) The elliptical environment
Ois more eccentric, with major axis 10 and minor axis 4. The resulting blue triangle is
more ”nonequilateral”, with length ratio 0.65 (after 1 iteration, and for 5 more iterations).
5 Conclusions and outlook
In this paper, we have explored the concept of mathematics as a universal language
capable of modeling the real world. By considering the interaction between an individual
(represented by a simplex) and its environment, we have gained a new perspective on
the dynamics of this relationship. Our findings have revealed that the environment not
only affects the individual, but the individual also has the potential to influence the
environment.
One of the main contributions of this research is the development of a framework for
iterations in arbitrary dimensions, where the simplex (or the individual) is influenced by
the environment. Through our investigations, we have shown that under the assumption
of the environment being a sphere around the simplex, convergence of the iteration can be
proven. Furthermore, we have observed that the limit simplex becomes regular, meaning
that the regularity of the environment is transferred to the individual. In particular, the
size of the environment directly affects the impact on the individual (compare Figure 4.3).
First numerical results imply that there is an optimal environment size maximizing the
impact on the individual. This phenomenon is also common in a social context. We are
influenced by the people close to us, but this influence is felt only up to a certain group
of people closest to us. Any more distant, and there is not enough connection to have an
impact.
Although we have provided numerical results demonstrating linear convergence rate for
most of the values of environmental radius, a rigorous theoretical proof is still pending.
The quest for such a proof remains an important direction for future research. Addition-
ally, we have identified several avenues for further exploration to broaden the scope of our
framework.
Some previously studied geometric transformations are known to give rise to anisotropy
[24]. The first step forward for the Chameleon transformation involves investigating the
13
O
(a)
O
(b)
O
(c)
O
(d)
Figure 5.2: Simultaneous projections method with initial triangle ∆0drawn in black and
four separate off-centered circular environments of radius 3. The environment centers are
(1,0), (0,1), (−1,0), (0,−1), respectively.
impact of anisotropic environments on the regularity of the simplex. Specifically, we are
interested in understanding how various shapes, such as rectangles, polygons, ellipses, or
arbitrary curves that surround the triangle, affect the convergence behavior. Similarly,
off-centered environments can be examined. This is a promising direction, as we see the
eccentricity of the environment affecting the ”nonequilarity” of the simplex in our pre-
liminary experiments (see Figure 5.1 and Figure 5.2). An interesting reverse question is:
given an initial triangle and the triangle it converges to, to what extent is the environ-
ment determined? In nature, it is often the case that environment does not determine
individual [3]. Is this also reflected in our Chameleon transformation? By extending our
analysis to anisotropic environments, we aim to uncover new insights into the interplay
between the individuum and its surroundings.
Expanding the framework to accommodate meshes of triangles represents another crucial
aspect of our future research. A regular mesh is important for convergence and accu-
racy in applications like the finite element method [12], transport theory [18][17][16], and
numerical partial differential equations with uncertain initial and boundary conditions
[14][5][15]. Although there has been recent work to make simulation accuracy indepen-
dent of mesh quality, these methods have high computation cost [19]. Existing mesh
smoothing methods include geometry-based [28], optimization [10] [6], physics-based [4],
deep learning [8], and hybrid methods [22]. Geometry-based mesh smoothing have seen
success in the past in many different types of meshes, including the well-known Laplace
smoothing [7], as well as other geometric element transforms [30] for tetrahedral [29] to
hexahedral [26] to mixed element [27] [25] meshes.
We acknowledge that the individuum-environment interaction becomes more intricate in
a mesh. Consequently, we need to address fundamental questions regarding the choice
of environment — e.g., whether it should be selected individually for each triangle or
as a global environment for the entire mesh. In the social context, a mesh of triangles
corresponds with having multiple individuals having to cooperate, which is affected by
their environment [11]. This idea of a mesh iteration where each triangle is an individual
trying to cooperate with the other triangles is reminiscent of game theory. Game theory
has been successfully applied to mesh smoothing in the past [23], and a similar game the-
oretic approach can be combined with the Chameleon transformation to mesh smoothing
in the future. The extension of the iteration process to work with multiple connected
triangles, either sequentially or simultaneously, requires careful consideration.
Furthermore, we recognize that our investigation has predominantly focused on the en-
14
vironment’s influence on the individuum. However, understanding the reverse process,
where the individuum affects the environment, holds significant potential. We have pro-
vided an initial example of such an iteration in Figure 5.3, but further exploration is
needed to grasp the extent of this bidirectional relationship. Unveiling the intricacies of
how the individuum shapes and modifies the environment would complete the picture and
foster a comprehensive understanding of the dynamics at play.
vE
vF
vH
vG
O
vA
vB
vC
vD
˜vA
˜vB
˜vC
˜vD
˜
˜vA
˜
˜vB
˜
˜vC
˜
˜vD
O
Figure 5.3: An example of a transformation, where an individuum affects the environment.
An object, in this case is a square E F GH, an environment Ois a quadrilateral ABCD.
On the picture two iterations of the Social transformation are shown.
In conclusion, this paper has shed light on the dynamic interplay between the individuum
and its environment in a mathematical framework. By demonstrating the transfer of
regularity from the environment to the individuum and presenting numerical evidence
of exponential convergence, we have established a solid foundation for future research.
The next steps involve investigating anisotropic environments, extending the framework
to meshes of triangles, and delving deeper into the reverse process of the individuum
affecting the environment. These endeavors will contribute to a more comprehensive
understanding of the mathematical modeling of real-world phenomena, paving the way
for novel applications and insights.
15
Declaration of interests
The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We would like to thank Dr. Michael Keckeisen, TWT, for his valuable discussions.
16
A Proofs
A.1 Original chameleon iteration
Theorem A.1. The sequence of iterations (2.3) converges in area, and in the limit area,
the triangle is equilateral.
Note: The area converges, but not the triangle. In the limit area, the corresponding
triangle is equilateral but will continue to rotate with more iterations.
Proof. As in the Section 2, consider the triangle ∆ := conv ({vA,vB,vC})∈R2. For
simplicity, we assume that vA,vB,vC∈ S1:= {x∈R2:∥x∥2= 1}. Define α, β, γ as
the central angles between vertices, such that
α= cos−1(⟨vA,vB⟩),
β= cos−1(⟨vB,vC⟩),
γ= cos−1(⟨vC,vA⟩).
Take a concentric second circle of radius R=√1 + ϵ(ϵ > 0) as the environment O. Let
one iteration take ∆ to ¯
∆ (circumradius √1 + ϵ), with λ, µ, ν such that
¯vA=vA+λ(vB−vA),
¯vB=vB+µ(vC−vB),
¯vC=vC+ν(vA−vC).
Define the areas of the triangles as vol2(∆) and vol2¯
∆. The equilateral triangle has the
uniquely maximum area among all triangles with a given circumradius. Thus, to show
the area of the triangle converges to that of an equilateral triangle, it is sufficient to show
that
vol2¯
∆/vol2(∆) ≥1 + ϵ,
with equality iff the original triangle is already equilateral.
We can see from Figure A.1 that ¯
∆ is composed of ∆ and three other triangles. Specifically,
vol2¯
∆= vol2(∆) + vol2(conv ({¯vA,vA,¯vC})) + vol2(conv ({¯vB,vB,¯vA})) +
+ vol2(conv ({¯vC,vC,¯vB})) .
The areas of the other three triangles are given by the ratio of the side lengths, so we
have
vol2¯
∆= vol2(∆) + λ(ν−1) ·vol2(∆) + µ(λ−1) ·vol2(∆) + ν(µ−1) ·vol2(∆)
Thus,
vol2¯
∆
vol2(∆) = 1 + λ(ν−1) + µ(λ−1) + ν(µ−1)
17
vA
vB
vC
¯vA
¯vB
¯vC
O
Figure A.1: One step of the iteration (2.3). The new triangle ¯
∆ consists of the original
triangle ∆ (gray) and three other triangles (red, green, blue).
Since ∥vA∥2=∥vB∥2=∥vC∥2= 1 and ∥¯vA∥2=∥¯vB∥2=∥¯vC∥2=√1 + ϵ, we have
three unknowns λ, µ, ν with the three equations
∥¯vA∥2
2=⟨vA+λ(vB−vA),vA+λ(vB−vA)⟩= 1 + ϵ,
∥¯vB∥2
2=⟨vB+µ(vC−vB),vB+µ(vC−vB)⟩= 1 + ϵ,
∥¯vC∥2
2=⟨vC+ν(vA−vC),vC+ν(vA−vC)⟩= 1 + ϵ.
Solving for λ, µ, ν we get
λ=pf(α) + 1
2,
µ=pf(β) + 1
2,
ν=pf(γ) + 1
2,
where we have defined fsuch that
f:x7→ ϵ
2(1 −cos x)
Plugging in these expressions for λ, µ, ν, our area ratio becomes
vol2¯
∆
vol2(∆) = 1 + pf(α) + 1
2pf(γ)−1
2+pf(β) + 1
2pf(α)−1
2+
+pf(γ) + 1
2pf(β)−1
2=1
4+pf(α)f(β) + pf(β)f(γ) + pf(γ)f(α),
where
f(x) = 1
4+ϵ
2(1 −cos x).
By AM-GM inequality, we have
vol2¯
∆
vol2(∆) ≥1
4+ 3 (f(α)f(β)f(γ)) 1
3
18
Now consider log f(x). Its derivative is
d
dx [log f(x)] = f′(x)
f(x)=−2ϵsin x
(1 + 2ϵ−cos x)(1 −cos x),
which is monotonically increasing on the interval (0,2π). Thus, log f(x) is strictly convex
for x∈(0,2π). This gives us
log [f(α)f(β)f(γ)]1
3=1
3[log f(α) + log f(β) + log f(γ)] ≥log fα+β+γ
3,
with equality iff α=β=γ. Since α+β+γ= 2π, we further have that
log [f(α)f(β)f(γ)]1
3≥log f2π
3= log 1
4+ϵ
3.
Since log is a monotonically increasing function, this gives
(f(α)f(β)f(γ))1
3≥1
4+ϵ
3.
Finally, we can plug this into our area ratio to get our desired result,
vol2¯
∆
vol2(∆) ≥1
4+ 3 1
4+ϵ
3= 1 + ϵ,
with equality iff the original triangle is equilateral. Thus, the sequence of areas of the
triangles converges to that of an equilateral triangle. Note that the transformation will ro-
tate any equilateral triangle, and thus the sequence of triangles converges to an equilateral
triangle only up to rotation.
19
B Supplementary materials
B.1 Numerical results for the 3D case
(a) (b)
Figure B.1: Example of the Chameleon iteration (2.4) for d= 3, O=R· S1,R= 1.3.
(a) An irregular tetrahedron inscribed in the unit sphere, vol3(∆0)/vol3ˆ
∆= 0.37. (b)
The transformed tetrahedron after one iteration of (3.3), vol3∆k/vol3ˆ
∆= 0.62.
20
HHHHH
H
k
R1.1 1.5 2 3 5 8 10
tkqktkqktkqktkqktkqktkqktkqk
0 8.8 ·10−10.33 8.8 ·10−10.69 8.8 ·10−10.81 8.8 ·10−10.89 8.8 ·10−11.03 8.8 ·10−11.26 8.8 ·10−11.38
1 4.5 ·10−10.82 1.9 ·10−10.97 9.8 ·10−20.99 4.3 ·10−21.02 2.4 ·10−21.10 2.0 ·10−21.19 2.0 ·10−21.11
2 3.3 ·10−10.91 6.5 ·10−20.99 1.6 ·10−21.00 2.9 ·10−31.02 6.0 ·10−41.07 1.8 ·10−40.81 1.1 ·10−40.77
3 2.4 ·10−10.95 2.2 ·10−21.00 2.7 ·10−31.00 1.9 ·10−41.02 1.0 ·10−51.01 6.3 ·10−71.01 3.5 ·10−71.31
4 1.8 ·10−10.97 7.4 ·10−31.00 4.3 ·10−41.00 1.2 ·10−51.02 1.3 ·10−70.91 6.5 ·10−91.18 4.1·10−19 0.78
5 1.4 ·10−10.98 2.5 ·10−31.00 7.1 ·10−51.00 6.7 ·10−71.01 1.6 ·10−90.96 6.1 ·10−11 0.83 1.2 ·10−11 -
6 1.1 ·10−10.99 8.6 ·10−41.00 1.2 ·10−51.00 3.7 ·10−81.01 3.0 ·10−11 1.05 2.5 ·10−13 - 1.3 ·10−13 -
7 8.2 ·10−20.99 2.9 ·10−41.00 1.9 ·10−61.00 2.0 ·10−91.00 6.9 ·10−13 - 3.0 ·10−15 - 4.4 ·10−16 -
8 6.3 ·10−20.99 9.9 ·10−51.00 3.1 ·10−71.00 1.0 ·10−10 0.99 1.6 ·10−14 - 4.4 ·10−16 - 0.0 -
9 4.9 ·10−21.00 3.4 ·10−51.00 5.0 ·10−81.00 5.3 ·10−12 0.99 0.0 - 4.4 ·10−16 - 0.0 -
10 3.8 ·10−21.00 1.2 ·10−51.00 8.0 ·10−91.00 2.8 ·10−13 - 0.0 - 4.4 ·10−16 - 0.0 -
11 2.9 ·10−21.00 4.0 ·10−61.00 1.3 ·10−91.00 1.5 ·10−14 - 0.0 - 4.4 ·10−16 - 0.0 -
12 2.2 ·10−21.00 1.4 ·10−61.00 2.1 ·10−10 1.00 8.8 ·10−16 - 0.0 - 4.4 ·10−16 - 0.0 -
13 1.7 ·10−21.00 4.6 ·10−71.00 3.3 ·10−11 1.00 0.0 - 0.0 - 0.0 - 0.0 -
14 1.3 ·10−21.00 1.6 ·10−71.00 5.2 ·10−12 1.00 0.0 - 4.4 ·10−16 - 0.0 - 0.0 -
Table 2: The values of tk(4.1) and convergence rate qk(4.2) with respect to iteration kand environmental radius R,d= 3.
21
Figure B.2: log10(vol3∆k+1−vol3∆k) with respect to the iteration kand environ-
mental radius R.
Figure B.3: Rate of convergence qwith respect to the environmental radius Rand iteration
k,d= 3.
22
References
[1] Alexander A Aksenov et al. (2022). “The molecular impact of life in an indoor
environment”. In: Science advances 8.25.
[2] Jacob E Allgeier et al. (2020). “Individual behavior drives ecosystem function and
the impacts of harvest”. In: Science Advances 6.9.
[3] Leander DL Anderegg (2023). “Why can’t we predict traits from the environment?”
In: New Phytologist 237.6, pp. 1998–2004.
[4] Raul Durand, BG Pantoja-Rosero, and Vicente Oliveira (2019). “A general mesh
smoothing method for finite elements”. In: Finite Elements in Analysis and Design
158, pp. 17–30.
[5] Jakob D¨urrw¨achter et al. (2020). “A hyperbolicity-preserving discontinuous stochas-
tic Galerkin scheme for uncertain hyperbolic systems of equations”. In: Journal of
Computational and Applied Mathematics 370, p. 112602.
[6] Bj¨orn Fabritius and Gavin Tabor (2016). “Improving the quality of finite vol-
ume meshes through genetic optimisation”. In: Engineering with Computers 32.3,
pp. 425–440.
[7] David A Field (1988). “Laplacian smoothing and Delaunay triangulations”. In:
Communications in applied numerical methods 4.6, pp. 709–712.
[8] Yufei Guo et al. (2021). “A new mesh smoothing method based on a neural network”.
In: Computational Mechanics, pp. 1–14.
[9] Tom´as Jim´enez (2017). The other side of assimilation: How immigrants are changing
American life. Univ of California Press.
[10] Jibum Kim (2012). Optimization-based meshing techniques for mesh quality im-
provement and deformation. The Pennsylvania State University.
[11] Maria Kleshnina et al. (2023). “The effect of environmental information on evolution
of cooperation in stochastic games”. In: Nature Communications 14.1.
[12] Daniel SH Lo (2014). Finite element mesh generation. CRC press.
[13] Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization. 2e. New
York, NY, USA: Springer.
[14] Louisa Schlachter and Florian Schneider (2018). “A hyperbolicity-preserving
stochastic Galerkin approximation for uncertain hyperbolic systems of equations”.
In: Journal of Computational Physics 375, pp. 80–98.
[15] Louisa Schlachter, Florian Schneider, and Oliver Kolb (2020). “Weighted essentially
non-oscillatory stochastic galerkin approximation for hyperbolic conservation laws”.
In: Journal of Computational Physics 419, p. 109663.
[16] Florian Schneider (2016). “Kershaw closures for linear transport equations in slab
geometry I: Model derivation”. In: Journal of Computational Physics 322, pp. 905–
919.
[17] Florian Schneider (2017). “Second-order mixed-moment model with differentiable
ansatz function in slab geometry”. In: arXiv preprint arXiv:1709.09032.
[18] Florian Schneider and Tobias Leibner (2020). “First-order continuous-and
discontinuous-Galerkin moment models for a linear kinetic equation: model deriva-
tion and realizability theory”. In: Journal of Computational Physics 416, p. 109547.
[19] Teseo Schneider et al. (2018). “Decoupling simulation accuracy from mesh quality”.
In: ACM transactions on graphics.
[20] Sai Sun and Rongjun Yu (2016). “Social conformity persists at least one day in
6-year-old children”. In: Scientific reports 6.1.
23
[21] Thomas Talhelm, Xuemin Zhang, and Shigehiro Oishi (2018). “Moving chairs in
Starbucks: Observational studies find rice-wheat cultural differences in daily life in
China”. In: Science advances 4.4.
[22] Dimitris Vartziotis, Doris Bohnet, and Benjamin Himpel (2018). “GETOpt
mesh smoothing: Putting GETMe in the framework of global optimization-based
schemes”. In: Finite Elements in Analysis and Design 147, pp. 10–20.
[23] Dimitris Vartziotis, Doris Bohnet, and Benjamin Himpel (2020). “Smoothing
Game”. In: arXiv preprint arXiv:2010.04956.
[24] Dimitris Vartziotis and Juri Merger (2018). “GETMe.anis: On geometric polygon
transformations leading to anisotropy”. In: arXiv preprint arXiv:1805.01767.
[25] Dimitris Vartziotis and Joachim Wipper (2009). “The geometric element transfor-
mation method for mixed mesh smoothing”. In: Engineering with Computers 25,
pp. 287–301.
[26] Dimitris Vartziotis and Joachim Wipper (2011). “A dual element based geometric
element transformation method for all-hexahedral mesh smoothing”. In: Computer
Methods in Applied Mechanics and Engineering 200.9-12, pp. 1186–1203.
[27] Dimitris Vartziotis and Joachim Wipper (2012). “Fast smoothing of mixed volume
meshes based on the effective geometric element transformation method”. In: Com-
puter methods in applied mechanics and engineering 201, pp. 65–81.
[28] Dimitris Vartziotis and Joachim Wipper (2018). The GETMe mesh smoothing
framework: A geometric way to quality finite element meshes. 1e. Boca Raton, FL,
USA: CRC Press.
[29] Dimitris Vartziotis, Joachim Wipper, and Bernd Schwald (2009). “The geometric
element transformation method for tetrahedral mesh smoothing”. In: Computer
Methods in Applied Mechanics and Engineering 199.1-4, pp. 169–182.
[30] Dimitris Vartziotis et al. (2008). “Mesh smoothing using the geometric element
transformation method”. In: Computer Methods in Applied Mechanics and Engi-
neering 197.45-48, pp. 3760–3767.
[31] Thierry Verdier and Yves Zenou (2017). “The role of social networks in cultural
assimilation”. In: Journal of Urban Economics 97, pp. 15–39.
[32] Jiayuan Xu et al. (2023). “Effects of urban living environments on mental health in
adults”. In: Nature Medicine, pp. 1–12.
[33] Elke Zeller et al. (2023). “Human adaptation to diverse biomes over the past 3
million years”. In: Science 380.6645, pp. 604–608.
24
