A discussion on the theory behind fractals, several different examples and applications of fractals in modern day life. This discusses the Coastline Paradox, Image Compression and uses within Creative Media
Fractals are the mathematical explanation of our world. Knowledge of fractals is essential to everyone's experience of their world. Here, I have explained the concept of fractals.
This presentation was created for a first year physics project at Imperial.
A presentation describing some of the applications of quantum entanglement, for example: quantum clocks, quantum computing, teleportation and quantum cryptography. Refers to specific experiment of teleportation carried out by NIST using time-bin encoding.
Quantum computation uses the quantistic physics principles to store and to process information on computational devices.
Presentation for a workshop during the event "SUPER, Salone delle Startup e Imprese Innovative"
ODSC India 2018: Topological space creation & Clustering at BigData scaleKuldeep Jiwani
Every data has an inherent natural geometry associated with it. We are generally influenced by how the world visually appears to us and apply the same flat Euclidean geometry to data. The data geometry could be curved, may have holes, distances cannot be defined in all cases. But if we still impose Euclidean geometry on it, then we may be distorting the data space and also destroying the information content inside it.
In the space of BigData world we have to regularly handle TBs of data and extract meaningful information from it. We have to apply many Unsupervised Machine Learning techniques to extract such information from the data. Two important steps in this process is building a topological space that captures the natural geometry of the data and then clustering in that topological space to obtain meaningful clusters.
This talk will walk through "Data Geometry" discovery techniques, first analytically and then via applied Machine learning methods. So that the listeners can take back, hands on techniques of discovering the real geometry of the data. The attendees will be presented with various BigData techniques along with showcasing Apache Spark code on how to build data geometry over massive data lakes.
Fractals are the mathematical explanation of our world. Knowledge of fractals is essential to everyone's experience of their world. Here, I have explained the concept of fractals.
This presentation was created for a first year physics project at Imperial.
A presentation describing some of the applications of quantum entanglement, for example: quantum clocks, quantum computing, teleportation and quantum cryptography. Refers to specific experiment of teleportation carried out by NIST using time-bin encoding.
Quantum computation uses the quantistic physics principles to store and to process information on computational devices.
Presentation for a workshop during the event "SUPER, Salone delle Startup e Imprese Innovative"
ODSC India 2018: Topological space creation & Clustering at BigData scaleKuldeep Jiwani
Every data has an inherent natural geometry associated with it. We are generally influenced by how the world visually appears to us and apply the same flat Euclidean geometry to data. The data geometry could be curved, may have holes, distances cannot be defined in all cases. But if we still impose Euclidean geometry on it, then we may be distorting the data space and also destroying the information content inside it.
In the space of BigData world we have to regularly handle TBs of data and extract meaningful information from it. We have to apply many Unsupervised Machine Learning techniques to extract such information from the data. Two important steps in this process is building a topological space that captures the natural geometry of the data and then clustering in that topological space to obtain meaningful clusters.
This talk will walk through "Data Geometry" discovery techniques, first analytically and then via applied Machine learning methods. So that the listeners can take back, hands on techniques of discovering the real geometry of the data. The attendees will be presented with various BigData techniques along with showcasing Apache Spark code on how to build data geometry over massive data lakes.
3D visualization today has ever-expanding applications in science, education, engineering, medicine, interactive multimedia like games, etc. Producers of graphics processing units (GPU) – are specialized electronic circuits designed to rapidly manipulate and alter computer memory in such a way so as to massively accelerate the visualization of 3D environments – bring ever faster products to the market every six months which is rapidly increasing the possibilities of near future visualization/simulation methods.
Neil Lambert - From D-branes to M-branesSEENET-MTP
Lecture by prof. dr Neil Lambert (Department of Mathematics, King’s College, London, UK & CERN, Geneva, Switzerland) on October 22, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
2. OVERVIEW
• Introduction
• Examples of Fractals
• Cantor Set
• Mandelbrot Set
• Sierpinksi Triangle
• Koch Snowflake
• Hausdorff Dimension
• Relation to Chaos Theory
• Area filling Fractals
• Applications of Fractals
• How to measure the length of a coastline
• Image compression
• Fractals and Stocks
• Use in creative media
4. WHAT ARE
FRACTALS
• Infinitely complex pattern
• Each part has the same
statistical character as the
whole.
• They are useful in modelling
things that have repeating
patterns
• Snowflakes
• Brownian Motion
• Biological systems
• Describe chaotic phenomena
• Crystal growth
• Galaxy Formation
6. FORMAL DEFINITION OF A FRACTAL
• The fractal dimension needs to exceed the topological dimension
• Topological dimension - coordinates needed to describe a point
• Fractal dimension - a measure of roughness
• Mandelbrot 1975
8. CANTOR SET
• Late 1800’s
• Infinite segments
𝑁𝑛 = 2𝑛
𝑁 = lim
𝑛 → ∞
𝑁𝑛 = ∞
• Zero Length
𝐿𝑛 =
2
3
𝑛
𝐿 = lim
𝑛→ ∞
𝐿𝑛 = 0
9. HAUSDORFF
DIMENSION
• One example of a fractal
dimension.
• Also called the similarity
dimension
• If after each step a fractal
makes 𝑚 copies of itself scaled
down by a factor of 𝑟 then we
have:
𝑚 = 𝑟𝑑
𝑑 =
ln 𝑚
ln 𝑟
10. CANTOR SET
• 𝑚 = 2 copies of the
previous step
• Scaled down by a
factor of 𝑟 = 3 so
𝑑 =
ln 2
ln 3
≈ 0.63
11. MANDELBROT SET
• Created from the logistics function:
𝑓 𝑧 = 𝑧2 + 𝑐
• The Mandelbrot set is the values for c for
which this function does not diverge
(after infinite steps) starting from 𝑧 = 0.
• Not perfectly self-similar
• Hausdorff Dimension of boundary
𝑑 = 2
12. SECTIONS OF THE
MANDELBROT SET
• Different areas have different
properties
• Main Cardioid body – set of
values c for which map has
attracting fixed point
• Period Bulb – values c for
attracting period 2 cycle
• Smaller bulb- larger cycle
13. JULIA SETS
• We defined the Mandelbrot by the logistics function. This is one
(very famous) example of a Julia Set
• The general definition is any complex, non-constant function f(z)
that is holomorphic and maps the Riemann sphere onto itself.
• This turns out to be all nonconstant complex rational functions
such that:
𝑓 𝑧 =
𝑝 𝑧
𝑞(𝑧)
where p(z) and q(z) are complex polynomials.
14. FUN JULIA
SETS
𝑓 𝑧 = 𝑧2 + 𝑐
• We know certain values c that can
generate interesting graphs.
−0.42 + 0.6𝑖
0.355 + 0.355𝑖
0.37 + 0.1𝑖
−0.54 + 0.54𝑖
• Obviously pictures are a little more
engaging
• We’ll show you where you can
generate your own later
18. KOCH SNOWFLAKE
• Finite Area – fits inside a circle
𝐴 =
8
5
𝑎0
• Same pattern on all sides
- Actually 3 fractals
- Focus on one side only
19. KOCH SNOWFLAKE
• Infinite Length 𝐿𝑛 =
4
3
𝑛
𝐿 = lim
𝑛→∞
𝐿𝑛 = ∞
• Look at one side – contains 4 copies scaled by 3
𝑑 =
ln 𝑚
ln 𝑟
≈ 1.26
20. RELATION TO CHAOS
THEORY
• Mandelbrot’s book - “Fractals and
Chaos the Mandelbrot Set and
Beyond”
• Mandelbrot set is linked to logistics
function.
• The areas “coloured-in” represent
stable periods. The end of the
cardioid is the split from one periodic
orbit to two.
• Chaos is present along the x-axis
afterwards.
27. HOW TO
MEASURE A
COASTLINE
• Should be easy?
• Length given as:
• 3,171km from the OSI
• 6,226km from Heritage Council
• 6,347km from US Defense
Mapping Agency
• Who is right?
• Depends on the line you draw
28. COASTLINE OF
BRITAIN
• Smaller the line the more accurate
the picture
• But no part is perfectly straight.
How small do you go?
• This is essentially a fractal
30. DIFFERENT METHODS
• There are a few different ways
• Manually
• Python
• We’ll start with the manual method. Programmes used were:
• Google Earth
• GIMP
• Mouse Clickr
34. VALUE OF THE DIMENSION
• This gives us a slope of m = -0.21 ± 0.06.
Our dimension is then:
• d = 1 - m = 1.21 ± 0.06
• Convert to grey image and then count the
boxes by the difference in two pixels. This
becomes dependent on the threshold you
set.
35. RESULT OF
GREY-SCALE
METHOD
• What we found was the result of our
dimension was entirely based on the
threshold we set. We plotted the result
against the threshold and obtained this
graph.
• Whilst this is interesting data, it shows that
we can’t be reliant on this image for getting
the dimension.
36. BASEMAP MODULE
• Allows you to plot areas of a map.
This is what we obtained with a
Mercator projection.
• Useful because it’s already black
and white. Threshold shouldn’t
matter anymore.
37. BASEMAP
MODULE
• If we plot the same thing
as before we get:
• Value obtained is 1.35
• Hutzler, S. (2013). Fractal
Ireland. Science Spin, 58, 19-20
• McCartney M., Abernethy G., and
Gault L. (2010). The Divider
Dimension of the Irish Coast. Irish
Geography, 43, 277-284.
40. IMAGE
COMPRESSION
• A way of reducing file size of digital
images
• Lossy v Lossless
• Fractal image compression involves
splitting an image into grids
• Create a function that goes from
compression to final image
• Associate small grids in the range
with larger grids in the domain
43. FRACTALS AND
THE STOCK
MARKET
• Efficient Market Hypothesis (EMH)
• Says that stocks always trade at their
fair value on exchanges
• Flaw is it doesn’t predict anomalies
• Fractal Market Hypothesis (FMH)
• Attempts to incorporate volatility and
chaos into the model
• In markets, stock prices act like
fractals. They appear to move in
replicating geometric patterns
• https://www.investopedia.com/terms/f/fractal-
markets-hypothesis-fmh.asp
44. CREATIVE
WORKS
• Have been used on Jackson Pollock’s
artwork to identify real works from
imitations with a 93% success rate.
• David Foster Wallace admitted that
the first draft of Infinite Jest was
based on a fractal design.
• Alexis Wajsbrot has made great use
of fractals in both Dr. Strange and
Mary Poppins Returns.
• Italian Mosaics 13th Century
• Medieval Churches
45. RECAP
General Fractals
• Cantor Set
• Mandelbrot Set
• Julia Sets
• Sierpinksi Triangle
• Koch Snowflake
Area Filling Fractals
• Peano Curve
• Hilbert Curve
Applications
• Coastline
• Image Compression
• Stock Trading
• Creative Works
46. IMAGE SOURCES
• Coastline
Lake Mead (ukga.com)
• Romanesco Broccoli
Scientific American - PDPhoto.org
• Fern
Wikipedia Commons - António Miguel de Campos
• Lightning
GeoGebra
• Mandelbrot Period
https://fractal.institute/hidden-numbers-and-basic-
mathematics-in-the-mandelbrot-set/
• Sierpinksi Antenna
V. A. Sankar Ponnapalli and P. V. Y. Jayasree
• Mandelbrot Chaos
Georg-Johann Lay. Public Domain
• Gorilla
Cincinnati Zoo
V. A. Sankar Ponnapalli and P. V. Y. Jayasree (March 31st 2018). Fractal Array Antennas and Applications, Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing, Kok Yeow You, IntechOpen
Take time on this slide and try to explain the concept slowly. Make sure to reinforce what was said before to allow people to make the connection.
Should be easy? Just draw a line.
Well not quite. It depends on who you ask.
Who’s right?
The answer is that is depends on how large of a line you draw and how far you zoom in.
We can see that when you vary the size of the line you get a more accurate reflection of the length of the coastline
The problem is that no stretch of coastline is going to be perfectly straight so you can continue doing this endlessly.
In this way, coastlines act like fractals and we can find the rate at which the length grows in comparison to the scale we are working at.
We need some figure to describe this rate.
There are a few different ways of achieving this. We can manually count the number of boxes on a picture with successively smaller grids or we can use python to automate some/all the work. We did both.
First let’s take a look at the manual method. This involved getting as high a res image from Google Earth as possible. It was then imported into GIMP and a grid was put over the image. The boxes were marked if the coastline crossed through them and a was used to count mouse clicks as we clicked through all the boxes.
Now let’s look at another way we can do it. We can convert an image to grey-scale and then count the boxes based on how different two pixels are in colour. This is dependent on the threshold you set. We used the same image for consistency and the converted it to grey like so:
But of course, there is a python module for everything. The one we utilised was the basemap module. It allows you to plot areas of a map. This is what we obtained with a Mercator projection.
What was useful about this is that we could set it to transparent and the “grey” version would just be either fully black or fully white. What we’d expect then is that the dimension we calculate is independent of the threshold
This is exactly what we observe. If we do the same method as before and plot the dimension against the threshold value, we see we get the same dimension value for each threshold.
This turns out to be 1.35 which is higher than what we had before but not too bad
There was a group in Trinity that did it the same manual method we did at got 1.20 and there was a group in University of Ulster who used dividers and got 1.23
Okay, so we have a value and it conforms with previous values. Well, what’s the point? Aside from the absolute riveting fun that is staring at maps for 2 hours, why do we do it.
The dimension is a measure of how rough a coastline is. We can compare this against other countries.
We can see the slope of the individual countries
West coast of Ireland v East Coast
South Africa smooth
Norway not very smooth
A lossy compression for digital images
Uses an iterated function system. Basically a set of functions that will create the image.
You separate the image into Ri blocks of size s x s
Then for each Ri search for blocks Di that are size 2s x 2s that are similar to Ri
Pick too little and it won’t look like your image. Pick too many and it’s computationally intensive
Now if people remember back to the first few lectures we were told to keep this entertaining so…
So, did you notice the Gorilla? Also, bonus marks for anyone that recognises this particular gorilla.
First thing we do is reduce the image by just averaging neighbouring pixels. This reduces the amount of computation that we need to do
The right hand image is the reduced image after being compressed and decompressed. We can see visual artificing on it. The way it’s being done is quite crude as we’re compressing and decompressing each individual RGB channel separately.
So if we convert the image to greyscale and then reduce it we can do the same process for different levels of compression and see how close it is to the original image.
The current “Main” market hypothesis is the efficient market hypothesis which says that stocks always trade at their fair value of exchanges
Gamestop and Bitcoin would probably beg to differ
Clearly this isn’t a perfect model but it’s main flaw is predicting anomalies
Alternative hypothesis is the fractal market hypothesis. This attempt to incorporate volatility and chaos into the model
In markets, stock prices act like fractals. They appear to move in replicating geometric patterns. The issue is you don’t know what time frame they’re repeating at.
There’s more information at that link.
Read
Clearly the ideas of fractals have been something that humans have found great fascination in for a lot of human history even if it’s only been mathematically rigorous for the past few decades
What we have found visually appealing is in fact mathematically complex and can used in a variety of situations.
We now invite people who have any questions and we’ll do a brief recap of what we’ve covered in case it jogs your mind.
