Hadron Collider Physics symposium 2012

11institutetext: Centro Studi e Ricerche e Museo Storico della Fisica âEnrico
Fermiâ, Rome, Italy 22institutetext: Sezione INFN, Bologna, Italy

# ALICE Physics Summary

Roberto Preghenella for the ALICE Collaboration 1122<EMAIL_ADDRESS>
###### Abstract

The ALICE experiment at the LHC has collected data in proton-proton (pp)
collisions at $\sqrt{s}$ = 0.9, 2.76, 7 and 8 TeV, in lead-lead (Pb–Pb)
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV and in proton-lead (p-Pb)
collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV. A summary overview of recent
experimental physics results obtained by ALICE is presented in this paper with
a selection of few representative measurements.

##  1 Introduction

ALICE (A Large Ion Collider Experiment) is a general-purpose heavy-ion
detector at the CERN LHC (Large Hadron Collider) whose design has been chosen
in order to fulfill the requirements to track and identify particles from very
low ($\sim$100 MeV/$c$) up to quite high ($\sim$100 GeV/$c$) transverse
momenta in an environment with large charged-particle multiplicities as in the
case of central lead-lead (Pb–Pb) collisions at extremely high energies.

;

Figure 1: Schematic layout of the ALICE detector with its main subsystems.

The ALICE experiment, shown in Figure 1, consists of a central-barrel detector
system and several forward detectors. The central system covers the mid-
rapidity region ($\left|\eta\right|\leq$ 0.9) over the full azimuthal angle
and it is installed inside a large solenoidal magnet with a moderate magnetic
of 0.5 T. It includes a six-layer high-resolution inner-tracking system (ITS),
a large-volume time-projection chamber TPC) and electron and charged-hadron
identification detectors which exploit transition-radiation (TRD) and time-of-
flight (TOF) techniques, respectively. Small-area detectors for high-$p_{\rm
T}$ particle-identification (HMPID), photon and neutral-meson measurements
(PHOS) and jet reconstruction (EMCal) complement the central barrel. The
large-rapidity systems include a single-arm muon spectrometer covering the
pseudorapidity range -4.0 $\leq\eta\leq$ -2.4 and several smaller detectors
(VZERO, TZERO, FMD, ZDC, and PMD) for triggering, multiplicity measurements
and centrality determination. A detailed description of the ALICE detector
layout and of its subsystems can be found in Aamodt:2008zz .

Since November 2009 when the first LHC collisions occured, the ALICE detector
collected proton-proton data at several centre-of-mass energies ($\sqrt{s}$ =
0.9, 2.76, 7 and 8 TeV). During the first two LHC heavy-ion runs, in the falls
of 2010 and 2011, the ALICE experiment recorded Pb–Pb collisions at a centre-
of-mass energy per nucleon pair of $\sqrt{s_{\rm NN}}$ = 2.76 TeV and could
profit of an integrated luminosity of about 10 $\mu\rm b^{-1}$ and 100 $\mu\rm
b^{-1}$, respectively. The instant luminosity exceeded $10^{26}\rm
cm^{-2}s^{-1}$ in the second run, higher than the design value. Proton-lead
(p–Pb) collision data were also recorded with the ALICE detector. This occured
during a short run performed in September 2012 in preparation for the main
p–Pb run scheduled at the beginning of 2013. Each beam contained 13 bunches; 8
pairs of bunches were colliding in the ALICE interaction region, providing a
luminosity of about $8\times 10^{25}\rm cm^{-2}s^{-1}$. Beam 1 consisted of
protons at 4 TeV energy circulating in the negative z-direction in the ALICE
laboratory system, while beam 2 consisted of fully stripped 208Pb ions at
$82\times 4$ TeV energy circulating in the positive z-direction. This
configuration resulted in collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV in the
nucleon-nucleon centre-of-mass system, which moves with a rapidity of $\Delta
y_{\rm NN}$ = 0.465 in the direction of the proton beam.

##  2 First results in proton-lead collisions

Particle production in proton-lead (p–Pb) collisions allows to study and
understand QCD at low parton fractional momentum $x$ and high gluon density.
Moreover it is expected to be sensitive to nuclear effects in the initial
state. For this reason p–Pb measurements provide an essential reference tool
to discriminate between initial and final state effects and they are crucial
for the studies and the understanding of deconfined matter created in nucleus-
nucleus collisions.

###  2.1 Charged-particle pseudorapidity density

Figure 2: Pseudorapidity density of charged particles measured in NSD p–Pb
collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV compared to theoretical
predictions.

The measurement of primary charged-particle pseudorapidity density performed
in non-single diffractive (NSD) p–Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02
TeV is reported in ALICE:2012xs . The resulting pseudorapidity density is
presented in Figure 2 as measured in the laboratory system for
$\left|\eta_{\rm lab}\right|<2$. A forward-backward asymmetry between the
proton and the lead hemispheres is clearly visible. The data is also compared
to several model predictions of particle production which have been shifted in
the laboratory system when needed. The comparison shows that the
pseudorapidity dependence is best described by the models DPMJET and HIJING
2.1 (where the gluon shadowing parameter $s_{g}$ was tuned on experimental
$\sqrt{s_{\rm NN}}$ = 200 GeV d–Au data at RHIC), whereas the saturation
models (KNL, rcBK, IP-Sat) exibit a steeper $\eta_{\rm lab}$ dependence. The
pseudorapidity density in the centre-of-mass system at mid-rapidity
$\left|\eta_{\rm cms}\right|<0.5$ is ${\rm d}N_{\rm ch}/{\rm d}\eta_{\rm
cms}=16.81\pm 0.71$, corresponding to $2.14\pm 0.17$ charged particles per
unit pseudorapidity per participant when scaled by the number of participating
nucleons, determined using the Glauber model Alver:2008aq .

###  2.2 Transverse momentum distribution

Figure 3: The nuclear modification factor of charged particles as a function
of transverse momentum in NSD p–Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02
TeV compared to measurements in central (0-5%) and peripheral (70-80%) Pb–Pb
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV.

The measurement of the transverse momentum $p_{\rm T}$ distributions of
charged particle in p–Pb collisions were also reported ALICE:2012mj . It was
previously shown that the production of charged hadrons in central Pb–Pb
collisions at the LHC is strongly suppressed Aamodt:2010jd ; :2012eq . The
suppression remains substantial up to 100 GeV/$c$ and is also seen in
reconstructed jets :2012is . Proton-lead collisions provide a control
experiment to establish whether the initial state of the colliding nuclei
plays a role in the observed high-$p_{\rm T}$ hadron production in Pb–Pb
collisions. In order to quantify nuclear effects, the $p_{\rm T}$-differential
yield relative to the proton-proton reference, the so-called nuclear
modification factor, is calculated. The nuclear modification factor is unity
for hard processes which are expected to exhibit binary collision scaling.
This has been recently confirmed in Pb–Pb collisions at the LHC by the
measurements of direct photon, Z0 and W± production, observables which are not
affected by hot QCD matter. In Figure 3 the measurement of the nuclear
modification factor in p–Pb collisions RpPb is compared to that in central
(0-5% centrality) and peripheral (70–80%) Pb–Pb collisions RPbPb. RpPb is
observed to be consistent with unity for transverse momenta higher that about
2 GeV/$c$. This demonstrates that the strong suppression observed in central
Pb–Pb collisions at the LHC is not due to an initial-state effect, but it is
rather a final state effect related to the hot matter created in high-energy
heavy-ion collisions.

##  3 Recent physics results

A limited selection of recent physics results obtained by the ALICE experiment
at the LHC is reported in the following section. They include both
measurements performed in proton-proton (pp) and in lead-lead (Pb–Pb)
collisions and are meant to give a feeling of the physics capabilities of the
experiment, though this cannot be done in a complete way in this report.

###  3.1 Exclusive $\rm J/\psi$ photoproduction

Figure 4: Measured differential cross section of coherent $\rm J/\psi$
photoproduction compared with models. The point at positive rapidity is the
reflected result from the measurement at negative rapidities.

Exclusive vector meson production in heavy-ion interactions is expected to
probe the nuclear gluon distribution for which there is considerable
uncertainty in the low-$x$ region and it has been studied so far in gold-gold
(Au–Au) collisions at RHIC. The first LHC results on exclusive photoproduction
of $\rm J/\psi$ vector mesons measured at forward rapidities in ultra-
peripheral Pb–Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has been
reported here Abelev:2012ba . The forward and mid-rapidity measured
differential cross section for coherent $\rm J/\psi$ production is compared
with calculations from various models in Figure 4. Best agreement is found
with models which include nuclear gluon shadowing consistent with the EPS09
parametrizations of the nuclear gluon distribution functions (AB-EPS09).

###  3.2 Direct photons

Figure 5: Direct-photon invariant yield in 0-40% central Pb–Pb collisions at
$\sqrt{s_{\rm NN}}$ = 2.76 TeV with NLO pQCD prediction and exponential fit in
low-pT region.

Direct-photon production has been measured by ALICE in proton-proton and in
Pb–Pb collisions Wilde:2012wc . The measurements when compared to NLD pQCD
predictions are found to be in good agreement for pp collisions and peripheral
Pb–Pb collisions (40-80% centrality). On the other hand, in the case of
central (0-40% centrality) Pb–Pb collisions the direct photon signal is well
reproduced by NLO pQCD only for photon momenta above 4 GeV/$c$, as shown in
Figure 5. The low-$p_{\rm T}$ excess, of about 20% at around 2 GeV/$c$, is
attributed to thermal photons, that is photons produced in the QGP phase by
the scattering of thermalized partons. The thermalized nature of the
production medium should be reflected in the $p_{\rm T}$ distribution of
thermal photons. The excess over NLO pQCD is fit in Figure 5 with an
exponential in the $p_{\rm T}$ range 0.8–2.2 GeV/$c$. The inverse slope of
this exponential is found to be T = (304 $\pm$ 51) MeV. In a similar analysis
performed in central (0-20%) Au–Au collisions at $\sqrt{s_{\rm NN}}$ = 200
GeV, the PHENIX experiment at RHIC measures an inverse slope parameter of T =
(221 $\pm$ 19 $\pm$ 19) MeV. The LHC value is about 40% higher than the one
measured at RHIC.

###  3.3 Light-flavour hadron production

Figure 6: Transverse momentum distributions of the sum of positive and
negative pions, kaons and protons for central Pb–Pb collisions. The results
are compared to RHIC data and hydrodynamic models. Figure 7: Mid-rapidity
particle ratios compared to RHIC results and predictions from thermal models
for central Pb–Pb collisions at the LHC.

Figure 8: $\rm K/\pi$ (top) and $\rm p/\pi$ production ratio in pp collisions
compared with PYTHIA Monte Carlo predictions and NLO calculations.

ALICE has measured the production yields of primary charged pions, kaon and
(anti)protons in a wide momentum range. Primary particles are defined as
prompt particles produced in the collision and all decay products, except
products from weak decay of strange particles. The measurements have been
performed both in proton-proton collisions at several centre-of-mass enegies
($\sqrt{s}$ = 0.9, 2.76 and 7 TeV) and in Pb–Pb collisions at $\sqrt{s_{\rm
NN}}$ = 2.76 TeV as a function of collision centrality.

In Pb–Pb collisions the $p_{\rm T}$ distributions and yields are compared to
previous results at RHIC and expectations from hydrodynamic and thermal
models. The results obtained for central Pb–Pb collisions are shown in Figure
6 and 7 and are also reported in :2012iu . The spectral shapes are harder than
those measured at RHIC, indicating a strong increase of the radial flow
velocity with the centre-of-mass energy. The radial flow at the LHC is found
to be about 10% higher than at RHIC energy. While the K/$\pi$ integrated
production ratio is measured to be in line with lower energy results and
predictions from the thermal model, both the p/$\pi$ and the $\Lambda/\pi$
ratios are lower than at RHIC and significantly lower (a factor $\sim$ 1.5)
than predictions. A possible explanation of these deviations from the thermal-
model predictions may be re-interactions in the hadronic phase due to large
cross sections for antibaryon-baryon annihilation Steinheimer:2012rd ;
Becattini:2012sq ; Pan:2012ne .

The $p_{\rm T}$-dependent production of charged kaons and protons normalized
to charged pions, respectively K/$\pi$ and p/$\pi$, are shown in Figure 8 for
pp collisions at $\sqrt{s}$ = 2.76 and 7 TeV where they are compared with
theoretical model predictions. No energy dependence is observed in the data
within the systematic uncertainties. The observed production ratios are not
reproduced by NLO calculations. PYTHIA Monte Carlo generator underpredicts the
proton-to-pion ratio at intermediate $p_{\rm T}$. Pion, kaon and (anti)proton
production in Pb–Pb collisions were compared to that of proton-proton
interactions and all show a suppression pattern which is similar to that of
inclusive charged hadrons at high momenta ($p_{\rm T}$ above $\simeq$ 10
GeV/$c$) OrtizVelasquez:2012te . This suggests that the dense medium formed in
Pb–Pb collisions does not affect the fragmentation. A similar conclusion can
be drawn by observing the proton-to-pion ratio measured in Pb–Pb collisions:
for intermediate momenta (3–7 GeV/$c$) it exibits a relatively strong
enhancement, a factor 3 higher than proton-proton collisions at $p_{\rm
T}\approx$ 3 GeV/$c$ and gets back to the proton-proton value at higher
momenta ($p_{\rm T}$ above $\simeq$ 10 GeV/$c$) OrtizVelasquez:2012te . A
similar observation is reported also for the $\rm\Lambda/K^{0}_{s}$ ratio and
possible explanations include among those proposed so far particle production
via quark recombination Fries:2003vb .

###  3.4 Heavy-flavour production

Figure 9: $p_{\rm T}$-differential invariant cross sections of electrons from
beauty hadron decays. The ratio of the data and the FONLL calculations are
shown in in the bottom panel and the dashed lines indicate the calculation
uncertainty. Figure 10: Average D meson suppression versus number of
participants compared with CMS non prompt $\rm J/\psi$ suppression.

ALICE has studied heavy-flavour production in pp and Pb–Pb collisions at the
LHC. Their prodution in proton-proton collisions is a tool to test pQCD
calculations in a new energy domain. The $p_{\rm T}$-differential production
cross section of prompt charmed mesons (D0, D+, D∗+, D${}^{+}_{s}$), heavy-
flavour decay electron and muons in pp collisions at $\sqrt{s}$ = 7 and 2.76
TeV are reported in Abelev:2012xe ; Abelev:2012pi ; Abelev:2012qh ;
ALICE:2011aa ; :2012sx ; :2012ik  and the measurements are well described by
pQCD prediction. The production cross section of electrons from semileptonic
decays of beauty hadrons was also measured at mid-rapidity
($\left|y\right|<0.8$) in pp collisions at $\sqrt{s}$ = 7 TeV :2012ic . The
results are shown in Figure 9 together with the pQCD FONLL prediction.

Heavy-flavuor production studies in Pb–Pb collisions at $\sqrt{s_{\rm NN}}$ =
2.76 TeV were also performed ALICE:2012ab ; delValle:2012qw . In particular,
the relative production of heavy-flavour particles with respect to that in
nucleon-nucleon interaction show a suppression of up to a factor of 5 at
$p_{\rm T}\sim$ 10 GeV/$c$ in central Pb–Pb collisions (0-7.5% centrality).
The suppression is observed to be similar for the three studied decay
channels, that is heavy-flavour electrons, heavy-flavour muons and prompt D
mesons. The average D meson production suppression is shown in Figure 10 as a
function of the number of participating nucleons and it is compared with the
measurement of non-prompt $\rm J/\psi$ suppression performed by CMS
CMS:2012vxa . The suppression of non-prompt $\rm J/\psi$ from B meson decays
reflects the in-medium energy loss of $b$ quarks. The two measurements provide
a first indication of different suppression for charm and beauty in central
collisions, that is of different in-medium energy loss.

###  3.5 Quarkonia

Figure 11: $\rm J/\psi$ production yield ${\rm d}N_{\rm J/\psi}/{\rm d}y$ as a
function of the charged particle multiplicity densities at mid-rapidity ${\rm
d}N_{\rm ch}/{\rm d}\eta$. Both values are normalized by the corresponding pp
minimum bias value. Figure 12: Inclusive $\rm J/\psi$ suppression $R_{\rm AA}$
versus the number of participant nucleons $N_{\rm part}$ in Pb–Pb collisions,
compared with results from PHENIX at RHIC.

ALICE has measured the inclusive $\rm J/\psi$ production in pp and Pb–Pb
collisions at the LHC, down to zero $p_{\rm T}$ Aamodt:2011gj ; Abelev:2012kr
. A unique study performed by ALICE in pp collisions is the determination of
the inclusive $\rm J/\psi$ production yield as a function of the charged
multiplicity at central rapidity. These results Abelev:2012rz  show a linear
increase of the yield with charged multiplcity which is not explained by
theory and model predictions yet. A similar increase is observed both a
forward and mid-rapidity, as shown in Figure 11.

Results on inclusive $\rm J/\psi$ production in Pb–Pb collisions clearly
indicate a saturation of the suppression at both central and forward rapidity
moving towards central collisions, a phenomenon not observed at lower energy
where the suppression increases as shown in Figure 12. There is in fact a
clear evidence for a smaller suppression at LHC with respect to RHIC energy
Abelev:2012rv . Differential studies of the suppression versus collision
centrality for various momentum bins seem to favour a scenario where
(re)combination processes play a sizeble role BraunMunzinger:2000px ;
Thews:2000rj . The observed hint for a non-zero anisotropic production signal
(elliptic flow) are in agreement with such a picture.

##  4 Summary

ALICE has obtained so far a wealth of physics results both from the analysis
of proton-proton collision data and from the first two LHC heavy-ion runs.
First results from a short pilot run with proton-lead have been already
obtained and the coming dedicated p-Pb run at the beginning of 2013 will set
the beginning of precision carachterization of the matter formed in heavy-ion
collisions at the LHC. A clear detector upgrade strategy plan for the LHC
luminosity upgrade has also been presented at this conference ken .

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45678 0 0 1111111.2222222  Exploratory parameter analysis in early prototype
Julia sets of quadratic fractals with Dual-Quaternions.

1 0

# Dual-Quaternion Julia Fractals

Ben Kenwright

###### Abstract

Fractals offer the ability to generate fascinating geometric shapes with all
sorts of unique characteristics (for instance, fractal geometry provides a
basis for modelling infinite detail found in nature). While fractals are non-
euclidean mathematical objects which possess an assortment of properties
(e.g., attractivity and symmetry), they are also able to be scaled down,
rotated, skewed and replicated in embedded contexts. Hence, many different
types of fractals have come into limelight since their origin discovery. One
particularly popular method for generating fractal geometry is using Julia
sets. Julia sets provide a straightforward and innovative method for
generating fractal geometry using an iterative computational modelling
algorithm. In this paper, we present a method that combines Julia sets with
dual-quaternion algebra. Dual-quaternions are an alluring principal with a
whole range interesting mathematical possibilities. Extending fractal Julia
sets to encompass dual-quaternions algebra provides us with a novel visualize
solution. We explain the method of fractals using the dual-quaternions in
combination with Julia sets. Our prototype implementation demonstrate an
efficient methods for rendering fractal geometry using dual-quaternion Julia
sets based upon an uncomplicated ray tracing algorithm. We show a number of
different experimental isosurface examples to demonstrate the viability of our
approach.

###### keywords:

graphics, dual-quaternion, visualization, fractals, geometry, graphical
processing unit

##  1 Introduction

### Geometric Properties of Complex & Dual Numbers

Complex numbers have a very elegant geometric interpretation. Specifically, we
can treat complex numbers as vectors in the complex plane. Addition and
subtraction then follow regular rules of vector arithmetic, while complex
multiplication can be seen as scaling the rotation of one vector by the
magnitude and argument of another, respectively. A dual number is a number of
the form $a+b\varepsilon$, where $a,b\in\mathbb{R}$ and $\varepsilon$ is a
non-real number with the property $\varepsilon^{2}=0$. Dual numbers are in
some way similar to complex numbers $a+bi$, where $i^{2}=-1$. You can imagine
dual numbers as situated on the orthogonal line centred at ‘a’ in the dual
number plane. When treating dual numbers as vectors the modulus is evaluated
as $||a+b\varepsilon||=abs(a)$ because of the nilpotency [Wang and Yu 2010].

### Dual-Quaternions

Dual-quaternion have been around since 1882 [Clifford 1882] but has gained
less attention compared to quaternions alone. Comparable to quaternions the
dual-quaternions have had a taboo associated with them, whereby individuals
avoid quaternion and hence dual-quaternions [Kenwright 2012]. While the
robotics community has started to adopt dual-quaternions in recent years, the
computer graphics community has not embraced them as whole-heartedly. Recent
work which has taken hold and has demonstrated the practicality of dual-
quaternions in computer graphics, some examples include: Kavan et al. [Kavan
et al. 2008] who demonstrated the advantages of dual-quaternions in character
skinning and blending. Frey and Herzeg [Frey and Herzeg 2011] extended Kavan
et al.’s [Kavan et al. 2008] work with dual-quaternions and qtangents as an
alternative method for representing rigid transforms instead of matrices.
Giving evidence that the results can be faster with accumulated
transformations of joints if the inferences per vertex are large enough. Selig
[Selig 2010] address key limitations in computer games by examining the
problem of solving the equations of motion in real-time by putting forward how
dual-quaternion give a very neat and succinct way for representing rigid-body
transformations. Vasilakis and Fudos [Vasilakis and Fudos 2009] discussed
skeleton-based rigid-skinning for character animation. Kuang et al. [Kuang et
al. 2011] presented a strategy for creating real-time animation of clothed
body movement.

### Algebra and Fractals

Julia set of a function $f$ is the set of all points $z$ in $C$ such that $f$
displays sensitive dependence at $z$. A popular Julia set function, and the
one we use in this paper, is of the form $f(z)=z^{2}+c$ for some constant $c$.
In the two dimensional complex domain, a Julia set is produced using the
iterative algorithm $z_{n+1}=z_{n}^{2}+c$, for a given complex number $c$. The
Julia set contains the initial complex values $z_{0}$ for which the system
converges. There have been a number of publications devoted to computational
and graphical aspects of the Julia sets [Ke and Panduranga 1990, Crane 2005].
including 3-dimensional and 4-dimensional contexts to generate fractal
geometry. In particular, modelling fractal geometries using quaternions is a
well researched topic in computer graphics [Pickover 1998]. Along similar
lines, Octonionic Julia sets have also been investigated in the literature
from the computational and graphical point of view. Dual-quaternions follow a
regular set of arithmetic rules (for performing common operations, such as,
multiplication, addition and magnitude - see Appendix A and Figure 1).
Extending the concepts developed for creating quaternion Julia sets, allows us
to formulate similar methods for dual-quaternion Julia sets.

### Contribution

The key contributions of this paper are: (1) process for building dual-
quaternion Julia sets; (2) graphical examples and experiments.

Figure 1: Dual-Quaternion \- Visual Overview of Quaternion and Dual-Quaternion
Components. To avoid confusion and enable the reader to easily distinguish a
quaternion from a dual-quaternion we use two discernible symbols to identify
them.

##  2 Related Work

### Quaternions to Dual-Quaternions

Clifford [Clifford 1882] published his intriguing work on dual-numbers in
1873, and provided us with a powerful tool for facilitating the analysis of
complex systems (e.g., mechanical, geometric) [Kenwright 2012]. The dual-
quaternion is an extension of dual-number theory whereby the numbers for the
dual-number equation are represented by quaternions. Remarkably, the dual-
quaternion algebra that results is very straightforward and elegant and
provides us an algebraically compact and efficient system for solving
otherwise complex problems [Kenwright 2012]. Dual numbers are pretty similar
to imaginary numbers but there is one important difference. With imaginary
numbers, $i^{2}=-1$, but with dual numbers, $\varepsilon^{2}=0$ (and
$\varepsilon$ is not 0), while this may seem like a small difference, it opens
up a whole interesting world of mathematical possibilities.

### Fractal Geometry

Generating a 4-dimensional version of complex numbers are called quaternions.
Alan Norton [Norton 1982] was the first to demonstrate the application of the
Quaternion Julia sets made by displaying a 3D ‘slice’ of the 4D space.
Quaternion Julet sets is actually a projection from 4 dimensions to 3
dimensions, akin to how a 2-dimensional square is a presentation of a
3-dimensional projection of a cube. Despite the added complexity, it possesses
an underlying smooth appearance (i.e., less interesting fractal detail when
you zoom in compared to the 2-dimensional version). As we show in this paper
higher-dimensional maths can be used to create 3-dimensional Juliet set
fractals. Daniel White’s ‘Mandelbulb’ [White 2007] takes a different approach.
He took the geometrical properties of the ‘complex plane’ where multiplication
becomes rotation and addition becomes movement of the plane in a particular
direction and applied them to a 3- dimensional space. The concept allows the
generation of striking visuals of how an apparently simple process can lead to
highly intricate sets. There are numbers of preceding studies about properties
of the fractals and Julia sets and how to implement them in computer graphics
[Crane 2005]. Similar to the existing results we ray-trace the interpolation
Julia set for making shading-reflecting like effect and apply the proposed
method into the stereographic projection of complex space to obtain a kind of
uneven and shaded surfaces.

Figure 2: Experimental \- Exploring visual possibilities through random
parameters. Parameter constant value from top left to bottom right:
c=(-0.10,0.8,-0.26,0.15)(-023,-0.38,-0.86,0.64);
c=(-0.67,-0.54,-0.07,0.02)(0.06,-0.53,0.15,-0.27);
c=(-0.98,0.27,0.40,0.20)(-0.37,-0.41,-0.24,-0.34);
c=(0.35,0.78,0.85,-0.57)(-0.22,0.06,-0.46,0.05);
c=(-0.17,-0.28,0.11,0.8)(0.06,0.44,-0.66,0.06);
c=(-0.04,0.95,0.4,-0.43)(0.09,-0.45,-0.27,-0.31).

##  3 Method

### Julia Set Fractals

Gaston Julia was the French mathematician who made the exciting discovery of
Julia Sets in 1918 [Mandelbrot 1983]. Julia sets are a popular fractal formed
using formula iterations. The Julia set is the union of all points $z$ in an
iterated complex function $f(z)$ that do not diverge as the number of
iterations of the function approach infinity (set of points form a ‘filled’ in
volume). Julia sets forms a boundary using the the filled-in set. There are a
large range of functions that encompass Julia sets, however, in this paper, we
employ the popular quadratic expression given in Equation 1

$f(z)=z_{n+1}=z_{n}^{2}+c$ (1)

where $z$ represents a variable of the form $a+bi$ ($a$ and $b$ are real-
numbers with $i$ an imaginary number). The value $c$ is also a complex number
and is constant for each unique Julia. Hence, as each value of $c$ gives a
different Julia set there are an infinite number of sets. In an uncomplicated
case, the value $a+bi$ represents a 2-dimensional complex plane with each
starting point $z_{0}$ for the iterative series representing a pixel position.
For each starting value there are two possibilities for what will happen,
either (1) as $n$ increases $f(z)$ will tend towards infinity or (2) the value
will stay within a bounded value. Points which do not stay within the bounding
limit are said to be in the escape set, while all other points are termed
prisoners (prisoner set). The common boundary between the escape set and the
prisoner set is called the Julia set (defined for a particular $c$).

### Quaternion Julia Sets

A quaternion is represented by two fundamental parts, a scalar real part ($w$)
and an imaginary vector part ($x,y,z$). We can extend the recursive Julia set
algorithm given in Equation 1 to use quaternions as given in Equation 3.
However, we need the squared quaternion, which is given in Equation 2. The
square of a quaternion is taken from previous quaternion geometric research
[Hart et al. 1989, Hart et al. 1990] with Julia sets (takes advantage of the
trigonometric properties of quaternions).

$\displaystyle q$ $\displaystyle=(\vec{v},w)=(x,y,z,w)$ (2) $\displaystyle
q^{2}$ $\displaystyle=(x^{2}-y^{2}-z^{2}-w^{2},\;2xy,\;2xz,\;2xw)$

The forward Julia iteration is given by Equation 3:

$q_{n+1}=q_{n}^{2}+c$ (3)

Figure 3: Level of Detail \- Trade-off between additional detail and
computational cost - number of sample points and maximum iterations (n) for
determining if a point is inside or outside the Julia set.

### Dual-Quaternion Julia Sets

Dual-quaternion Julia sets is the same as ordinary Julia sets except that
dual-quaternions are used instead of complex numbers. Generation of the
fractal , the function $\zeta_{n+1}=f(\zeta)$ is iterated and if the series is
bounded when $n$ tends to infinity then the dual-quaternion $\zeta_{0}$
belongs to the Julia set.

$\displaystyle\zeta$ $\displaystyle=q_{a}+q_{b}\varepsilon$ (4)
$\displaystyle\zeta^{2}$ $\displaystyle={q_{a}^{2}+q_{b}^{2}\varepsilon}$

where $a$ and $b$ are the dual-quaternion parameters (i.e., real and dual
number represented as two quaternions). We apply Equation 2 to calculate the
‘squaring’ of each component and add the dual-quaternion constant to achieve
each forward iteration given by the Julia Equation 5.

$\zeta_{n+1}=\zeta_{n}^{2}+c$ (5)

where $\zeta$ and $c$ are dual-quaternions; $n$ is the number of iterations,
typically between $6$ and $15$ (Figure Dual-Quaternion Julia Fractals shows
the impact of modifying the number of iterations on the level of detail).

### Practical Considerations

Impossible to iterate each sample point to infinity - hence, a suitable
maximum iteration range is defined. If the series has not reached the escape
limit by then, the point is considered inside the set. In practice, a fairly
small number of iterations gives reasonably good results [Hart et al. 1989]
(e.g., less than 10). However, the more iterations the more accurate the
fractal surface. As shown in Figure 3, the more iterations and the finer the
resolution provide additional detail for the geometric model at increased
computational cost.

### Dimension Reduction

Dual-quaternion Julia sets are in 8-dimensional space. How to visualize the
dual-quaternion Julia set in 3-dimensional space. We need to map the
8-dimensions into 3-dimensional space. There are two approaches: (1) render a
single 3-dimensional slice of the dual-quaternion space or (2) project the
dual-quaternion space onto 3-dimensions to see a 3-dimensional shadow of the
Julia set. The approach used in this paper is to intersect the dual-quaternion
space with a plane; essentially making five of the dual-quaternion components
(dimensions) constant values.

### Ray-Marching

Reduce the computational cost of testing every point in the 3-dimensional
space. Slow to generate a model of acceptable granularity in a reasonable
space of time. For example, see comparisons with a Voxel-based render shown in
Figure 5. To offer a fast and efficient visualization solution, we utilize a
ray-marching algorithm. The ray-marching algorithm is able to produce high-
quality detailed renders of the dual-quaternion Julia set at various levels.
We employ the distance estimation method for the Julia set [Dang et al. 2002].
This distance estimator can be used to accelerate the ray tracing process
using unbounding volumes. The estimation gives a lower-bound on the distance
to the fractal from any given point outside the fractal. While the Equation 6
provided an initial approximation, we found the calculation from Hart et al.
[Hart et al. 1989] given in Equation 7 provided more aesthetically pleasing
results (and required less tweaking/tuning).

$\displaystyle
d\geq\alpha\left(\frac{|\zeta_{n}|}{|\zeta_{n}^{\prime}|}\right)$ (6)
$\displaystyle|\zeta_{n}^{\prime}|=2|\zeta_{n+1}||\zeta_{n+1}^{\prime}|$

where $\alpha$ is analogous to the grid size and typically has a small value
(e.g., $\leq 0.1$). Once the lower-bound on the distance is known, we are able
to step along the ray until we hit the fractal or have stepped to many times
without hitting the fractal. An additional optimization is to have rays
intersect with a sphere encapsulating the fractal first. This helps speed up
the ray tracing algorithm to quickly discard rays that would not intersect the
fractal.

$\displaystyle d\geq
0.5\frac{|\zeta_{n}|}{|\zeta_{n}^{\prime}|}log(|\zeta_{n}|)$ (7)

Figure 4: High-Detail \- Emphasising detail within the dual-quaternion Julia
set geometry by using a larger number of iterations. c =
(-0.04,0.95,0.4,-0.43)(0.09,-0.35,-0.27,-0.31), n=15.

### Lighting/Illumination

Shading is an extremely important process in creating a realistic and
aesthetically pleasing image. As we are rendering a 3-dimensional model,
lighting is crucial to visually identify differences in geometric detail. We
employ the Phong model as it is straightforward and easy to incorporate into
the ray-tracing algorithm. However, the Phong model requires the surface
normals. We calculate a rough surface normal by sampling the surrounding
points (i.e., above, below, left and right). When a point in the Julia set is
found the four neighbouring points are also tested to see if they are inside
or outside the set. These points are required for generating the neighbouring
surface that allows us to compute the surface normal ($\vec{n}$). We calculate
the divergence at each pixel by iterating several points with a small delta.
While the slope at any point on the surface of the Julia set is undefined,
closer examination of a points surroundings, for example scanned at higher
iteration and z-resolution, would reveal a different environment. An exact
normal can not be calculated, however the approximation using a sample set of
the surrounding points provides an acceptable surface normal for our
experiments. The Phone illumination calculation for a single light is given
below in Equation 8.

$I=k_{a}I_{a}+k_{d}I_{d}(\vec{n}\cdot\vec{l})+k_{s}(\vec{r}\cdot\vec{v})^{s}$
(8)

where $I$ is the total illumination, $k$ is the attenuation coefficient for
the object material (ambient, specular and diffuse), $\vec{l}$ is the vector
to the light, $\vec{r}$ is the reflected light vector (i.e.,
$R=2(\vec{n}(\vec{n}\cdot\vec{l}))-\vec{l}$ ), $\vec{n}$ is the normal vector
at the object surface, $\vec{v}$ is the vector from the eye point, $s$ is the
specular exponent, and $I_{a},I_{s},I_{d}$ are the light intensities.

Figure 5: Voxel vs Ray-Traced \- Initial prototypes tried to utilize Voxels on
the GPU [Zadick et al. 2016] (i.e., geometry shader) - to create a real-time
prototyping tool for experimentation. However, a ray-tracing algorithm was
more suitable for emphasising detail due to computational/resolution limits.
Top ray-traced image, middle $50\times 50\times 50$ voxel grid and bottom
$100\times 100\times 100$ voxel grid.

### Graphical Processing Unit (GPU)

Fractal rendering has a high computational costs and can be parallelised
easily. For example, Keenan [Crane 2005] demonstrated the generation of
quaternion Julia sets using a GPU for the ray-tracing algorithm (fragment
shader). We employ a similar approach with the ray-marching algorithm on the
GPU used to render the geometric surface of the dual-quaternion Julia set.

##  4 Experimental Results

### Parameter Tuning/Experimenting

Figure Dual-Quaternion Julia Fractals and Figure 6 show some preliminary
experimental results through parameter tuning/changing. As shown in Figure 3,
adjusting the maximum number of iterations and the in/out Julia set range
allows us to capture additional low-level details (i.e., at additional
computational cost and the moire effect due to the problem of too much detail
in the image/model). The simulations provide a strong correlation in style and
appearance to quaternion Julia sets (twirling candy floss appearance). While
the dual-quaternion space has a clearly defined sum, product and norm - the
distance squared estimator in this paper uses a quaternion calculation for the
real and dual components which means it still has a strong coupling to a
4-dimensional space. Of course, this leaves the door open for further
exploratory work into other distance square estimator variations using dual-
quaternion algebra.

##  5 Conclusion/Discussion

In conclusion, fractals are a powerful tool for generating interesting
patterns. Extending the Julia set to higher dimensions allows us to form
shapes with strong stereoscopic properties which exhibit intertwisting,
irregular fringes and transformations. In this paper, we employed dual-
quaternion algebra to realize a 3-dimensional fractal using the Julia set.
When trying to capture the dimensional properties of dual-quaternions using
Julia sets, we rendered snapshots from different sample points. There is still
lot to be discovered about Julia sets in the dual-quaternion space. The rules
defined by dual-quaternion algebra are applicable to Julia set to extend the
concept to multiple dimensions. Techniques for visualizing the higher
dimensional Julia set as a three dimensional object. We describe our attempt
to extend the already established technique of ray tracing Julia sets to
incorporate dual-quaternion properties. We also discussed optimization
algorithm for reducing computational times through exploitation of the
graphical processing unit (allowing for exploratory prototyping). The
scientific value of these images can be questioned, but they possess aesthetic
factors of fascination.

Figure 6: Early Experiments \- Poorly defined distance parameters - exploring
visual possibilities through random values.

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Figure 7: High-Detail \- Emphasising detail within the dual-quaternion Julia
set geometry by using a larger number of iterations. c =
(-0.39054,-0.58679,0,0)(0.0,0.5632,0,0.05), n=15.

##  Appendix A Appendix

### Quaternion Operations

The quaternions were discovered by Hamilton in 1843 as a method of performing
3-D multiplication [Kenwright 2012]. A quaternion $q$ is given by Equation 9.
Since we are combining quaternions with dual number theory, we give the
elementary quaternion arithmetic operations.

$q=[s,\vec{v}],(q_{w},q_{x},q_{y},q_{z})$ (9)

where $s$ scalar part is $s=q_{w}$ and vector part is
$\vec{v}=(q_{x},q_{y},q_{z})$. The four-tuple of independent real values
assigned to one real axis and three orthonormal imaginary axes: $i,j,k$.

  * •

addition:
$q_{1}+q_{2}=[s_{1},\vec{v}_{1}]+[s_{2},\vec{v}_{2}]=[s_{1}+s_{2},\vec{v}_{1}+\vec{v}_{2}]$

  * •

additive identity: $0=[0,0]$

  * •

scalar multiplication: $kq=[ks,k\vec{v}]$

  * •

multiplication:
$q_{1}q_{2}=[s_{1},\vec{v}_{1}][s_{2},\vec{v}_{2}]=[s_{1}s_{2}-\vec{v}_{1}\cdot\vec{v}_{2},s_{1}\vec{v}_{2}+s_{2}\vec{v}_{1}+\vec{v}_{1}\times\vec{v}_{2}]$

  * •

multiplication identity: $1=[1,0]$

  * •

dot product: $q_{1}\cdot
q_{2}=(q_{1x}q_{2x}+q_{1y}q_{2y}+q_{1z}q_{2z}+q_{1w}q_{2w})$

  * •

magnitude: $||q||=\sqrt{(}s^{2}+||\vec{v}||^{2})$

  * •

conjugate $q^{*}=[s,-\vec{v}]$

### Dual-Quaternion Operations

The elementary arithmetic operations necessary for us to use dual-quaternions.

  * •

dual-quaternion: $\zeta=q_{r}+q_{d}\varepsilon$

  * •

scalar multiplication: $s\zeta=sq_{r}+sq_{d}\varepsilon$

  * •

addition: $\zeta_{1}+\zeta_{2}=q_{r1}+q_{r1}+(q_{d1}+q_{d2})\varepsilon$

  * •

multiplication:
$\zeta_{1}\zeta_{2}=q_{r1}+q_{r2}+(q_{r1}q_{d2}+q_{d1}q_{r2})\varepsilon$

  * •

conjugate: $\zeta^{*}=q_{r}^{*}+q_{d}^{*}\varepsilon$

  * •

magnitude: $||\zeta||=\zeta\zeta*$

where $q_{r}$ and $q_{d}$ indicate the real and dual part of a dual-
quaternion.