But, as with any software, there are some problems that you might face when you're trying to install it on your machine
Installation problems
When I was trying to install flutter on my machine, I faced some problems that I want to share with you.
First of all; before installation, you have to have:
- Git
- Powershell or a cmdline software.
- Minimum 7gb of disk space
After you have these, you can start the installation process.
First, you have to clone the flutter repository from the github. You can do this by typing the following command in your terminal:
git clone https://github.com/flutter/flutter.git
After you have cloned the repository, you have to add the flutter/bin directory to your PATH variable. Now this step is where I have stuck earlier when installing flutter, as I didn't realize I had to add the bin folder, so I only added flutter folder which unsurprisingly it did not work
After you have added the bin folder to your PATH variable, you can run the following command to check if the flutter is installed correctly:
flutter doctor
flutter doctor --android-licenses
After you have ran this command, you will see the output of the flutter doctor command. Now the next step involves downloading and installing Android Studio which is a straightforward process. Even if you don't use the Android Studio which is powered by Jetbrains IDE, you have to install it because it is needed to download android toolchain
After you have installed the Android Studio, you have to open it and go to the settings. In the settings, you have to go to languages section where there is an Android SDK section where you can download the Android Command Line Tools.
After you have downloaded the tools, you can close the Android Studio and go back to the IDE of your choosing and start developing awesome apps!
]]>Stateless Widgets
A StatelessWidget is immutable, meaning it cannot change its state after being created. It is designed for widgets whose configuration and appearance remain constant throughout their lifecycle.
Characteristic properties of Stateless Widgets
- Immutable: Once created, the widget cannot change.
- Lightweight: Suitable for UI elements that do not need to update or interact dynamically.
- Rebuild Mechanism: When the widget tree is rebuilt (e.g., due to a parent widget's state change), a new instance of the StatelessWidget is created.
Stateless widgets are typically used in Static text or images, icons and layout containers such as Row, Column or Containers. Example code given below:
class MyStatelessWidget extends StatelessWidget {
final String title;
MyStatelessWidget({required this.title});
@override
Widget build(BuildContext context) {
// Displays the title passed during creation
return Text(title);
}
}
In this example, MyStatelessWidget simply displays the title. You can't update the title after the widget is created.
Stateful Widgets
A StatefulWidget is dynamic and can change during its lifecycle based on user interaction, data updates, or other events. It consists of two classes:
- StatefulWidget Class: Represents the widget itself.
- State Class: Maintains the state of the widget and contains logic to update the UI.
Characteristic properties of Stateful Widgets
- Dynamic State: Can update or modify the UI in response to events.
- State Management: The State class stores properties that change and provides methods like setState to trigger updates.
- Lifecycle Methods: Includes methods such as initState, setState, and dispose to manage the widget's lifecycle.
Stateful Widgets are typically used in Form inputs, Buttons with toggling states, Animations or timers or generally Widgets that respond to asynchronous data(Such as API calls).
class MyStatefulWidget extends StatefulWidget {
@override
_MyStatefulWidgetState createState() => _MyStatefulWidgetState();
}
class _MyStatefulWidgetState extends State {
int counter = 0;
void incrementCounter() {
setState(() {
counter++; // Updates the counter and refreshes the UI
});
}
@override
Widget build(BuildContext context) {
return Column(
mainAxisAlignment: MainAxisAlignment.center,
children: [
Text('Counter: $counter'),
ElevatedButton(
onPressed: incrementCounter,
child: Text('Increment'),
),
],
);
}
}
In this example, pressing the button updates the counter state, which triggers a UI rebuild to reflect the new value. This is also the default code when building a new project in flutter.
Choosing Between Stateless and Stateful Widgets
- Use StatelessWidget when the widget does not need to change after being created.
- Use StatefulWidget when the widget needs to respond to user input, animations or data updates dynamically.
Understanding the difference between Stateless and Stateless Widget is crucial as it affects the design of the whole application!
]]>
Volatile memory is stored in a RAM modules. (Photo by Possessed
Photography on Unsplash)
Manual memory management
Typically in older programming languages for example, in C language you manually get a memory space with certain functions. So to get a memory of a certain quantity, you have to type how many bytes you want to allocate and free that memory space once you're done using it. To give an example of this, here is an example code written in C:
C
#include <stdio.h>
#include <stdlib.h>
int main() {
int *ptr;
int n = 5;
ptr = (int*) malloc(n * sizeof(int)); // allocate memory for 5 integers
if (ptr == NULL) {
printf("Memory allocation failed");
exit(1); // exit the program with an error code
}
// assign values to the memory block
for (int i = 0; i < n; i++) {
*(ptr + i) = i;
}
// print the values from the memory block
for (int i = 0; i < n; i++) {
printf("%d ", *(ptr + i));
}
free(ptr); // free the allocated memory
return 0;
}
The Malloc function takes one argument, which is the amount of memory to allocate in bytes. It then returns a pointer to the allocated block of memory. In the case of allocation, the operation is not successful, the function returns a NULL pointer. Here in the given code we've allocated 5 integer amounts of bytes to ptr and assigned values to them. Then finally we freed the allocated memory using the free function. Freeing existing memory at the last step is really important in order to prevent memory leaks. The common error people make when building large-scale codebases on these languages is that when at any point of the program, the allocated memory is not freed, the computer won't take it back because it is flagged as being used by that program. The other common mistake is when you freed the memory and you try to access that same bit of memory, in that case, the OS might have given that block to some other process and because of that, you might get compile-time errors or some random value you didn't ask for.
Photo by Denny
Müller on Unsplash
In this approach, as we have seen from the examples, we might encounter loads of problems. So in order to get ahead of that, most high-level programming languages have some sort of garbage collection built in. In short, garbage collection basically gives you memory when you don't need it anymore.
Ways of garbage collection
The most obvious way of garbage collection is reference counting. In this method, the allocated memory has a property called reference counter, you keep track of how many references that bit of memory has, and you free the aforementioned memory when it finally has zero references in the program. This approach has its own tradeoffs, firstly you have to remember to add or decrease this property and the other one is that with this approach you now have an integer associated with all of the memory spaces in your program which makes the program memory inefficient and can decrease the program's performance. Another problem is some sort of memory leak where you have some memory blocks with reference counter that doesn't go zero in order to free them. This in itself can be classified as a memory leak. With its tradeoffs, the reference counting is used in programming languages like Python where it's used with other sound methods of garbage collection like mark and sweep algorithm.
Mark and sweep
To understand the mark and sweep algorithm we need to know about live memory and dead memory concepts. Live memory is a memory block that is actively referenced, whereas dead memory is a memory block that doesn't have any references to it.
In this example, x and y is referencing memory blocks 1 and 2 so they are live memory and
therefore marked.
Mark stage goes through all of the accessible memory blocks, even the blocks that are not directly referenced. Here in this example if we remove the y reference to the memory block 2:
Marking goes through all the direct or indirect references marks them one by one sort of like a boolean value, because of that the program size is lower than the reference count method.
Sweeping stage frees up the unmarked memory block, making our program less dependent on high chunks of memory.
Source section
- Debnath, M. (2023) Understanding garbage collection in go, Developer.com. Available at: https://www.developer.com/languages/garbage-collection-go/ (Accessed: May 3, 2023).
- DeBrie, A. (2023) Python garbage collection: What it is and how it works, Stackify. Available at: https://stackify.com/python-garbage-collection/ (Accessed: May 3, 2023).
- Garbage collection in Java (2022) GeeksforGeeks. GeeksforGeeks. Available at: https://www.geeksforgeeks.org/garbage-collection-java/ (Accessed: May 3, 2023).
- Garbage Collector Design (no date) Python Developer's Guide. Available at: https://devguide.python.org/internals/garbage-collector/ (Accessed: May 3, 2023).
- Heller, M. (2023) What is garbage collection? Automated Memory Management for your programs, InfoWorld. InfoWorld. Available at: https://www.infoworld.com/article/3685493/what-is-garbage-collection-automated-memory-management-for-your-programs.html (Accessed: May 3, 2023).
:max_bytes(150000):strip_icc():format(webp)/nietzscheportrait-a6cfe34a104349e5b59991a1c94ea482.jpg)
Nietzsche emphasised will and power in his works.
In 1864 he entered the University of Bonn and shortly after his entry, he focused his academic life on Philology where he wanted to study classical languages. His contribution and curiosity towards ancient texts (mainly ancient Greek) drew the attention of Professor Ritschl and he was praised highly of him during that period. With all the praise he became semi-famous in the academic scene. He, later on, entered the University of Basel in Switzerland and became the Professor of Greek Language and Literature just at 24 years old.
Depiction of Basel in 1493.
But he quickly lost interest in becoming an academician and later on moved to Sils Maria on the outskirts of the Swiss Alps and lived relatively peacefully while working on his sensational philosophic ideas. He published The Birth of Tragedy, Human All to Human, The Gay Science, Thus Spoke Zarathustra, and many more where he talked in great depth about his philosophy. At that point in his life, he was having difficulties with his family, was failing relationships with women and his books weren't selling as expected. And when he was mid 40's he had a mental breakdown and when he was trailing on the street he saw a horse getting beaten by its driver and quickly embraced the horse shouting "I understand you". Nietzsche never recovered from that breakdown.
Nietzsche never really recovered from his mental breakdown.
Despite being depressed and having constant breakdowns, Nietzsche's philosophy was full of heroism and self-improvement. His ideal type of human was called an ÜBERMENSCH(superman) who overcomes difficulties and accepts what life throws at them. His work emphasized on getting people to become who they are in life. In order to achieve that, we need to implement 4 main recommendations.
Four main recommendations of Nietzsche
The first one of the recommendations is owning up to envy whereas major Abrahamic religions teach envy as something to be feel ashamed of and it is associated with evil. However, to Nietzsche, being envious as long as we utilize it to make our life better than today. Every person should be seen as a goal rather than a pure object that makes us jealous. People who cross the boundaries of success are hinting us who we might become one day. This approach basically tells us that we should face up our true desires and resist them and only then if we encounter failure we should mourn it. Übermensch utilizes that principle in his/her life.
Nietzsche's idea of superman was not as hollywood depicted but was much more subtle and
interesting man.(Photo by Mehdi
MeSSrro on Unsplash
)
His second recommendation was "Don't be a Christian". In Abrahamic religions, envy is associated with evil and emerged from the mindset of timid slaves that lacked the courage to get what they wanted. Those wanted entities could be money, fame, power, and women. Being a Christian, one was always clung to a philosophy that embraces cowardice. This was called slave morality in Nietzsche's terms. Religious people deemed these values as evil and embraced a hypocritical ideology and condemned what they always wanted. So in other words, sexlessness, and poverty became some sort of purity and goodness. Therefore Christianity is a giant cage for people who are in resentful denial.
To Nietzsche being a christian is accepting a hypocritical creed.
The third one on the list is "Never drink alcohol." Nietzsche himself was quite an interesting man as he only drank water. He makes the argument that narcotic substances and mind-numbing ideologies are the most dangerous to modern civilizations as it impedes the thinking process and numb people physiologically and mentally. Nietzsche hated the idea of numbing pain and reassurance of ideas at the expense of facing the truths and changing our lives for the better. As you can tell by these recommendations, he was an anti-conformist from the heart. As he calls it "The secret of a fulfilled life is: live dangerously."
"Taking action and facing problems head-on is the correct approach to life."
The last recommendation is "God is dead". His statement is not some sort of a celebration call, and the decrease in belief in higher order is not anything to cheer about. He understood that religion is false however, he also acknowledged that the belief helps us cope with the distress of life. This kind of deterioration of belief should be replaced by culture and fine arts in the form of art, music etc. For Nietzsche, philosophers should be on the fore front in handling great developments in social life and guide people when transitioning from old to the new age.
Source section
- Magnus, B. (2023) Friedrich Nietzsche, Encyclopædia Britannica. Encyclopædia Britannica, inc. Available at: https://www.britannica.com/biography/Friedrich-Nietzsche (Accessed: April 25, 2023).
- Mambrol, N. (2019) The philosophy of Friedrich Nietzsche, Literary Theory and Criticism. Available at: https://literariness.org/2019/04/18/the-philosophy-of-friedrich-nietzsche/ (Accessed: April 25, 2023).
- May-Hobbs, M. (2023) Nietzsche: A guide to his most famous works and ideas, TheCollector. Available at: https://www.thecollector.com/nietzsche-famous-works-and-ideas/ (Accessed: April 25, 2023).
- Rothfeld, B. (2018) How to live better, according to Nietzsche, The Atlantic. Atlantic Media Company. Available at: https://www.theatlantic.com/magazine/archive/2018/10/nietzsches-guide-to-better-living/568375/ (Accessed: April 25, 2023).
- Töniges, S. (2020) Nietzsche: The thinker who exploded the philosophy world – DW – 08/25/2020, dw.com. Deutsche Welle. Available at: https://www.dw.com/en/friedrich-nietzsche-the-dynamite-german-philosopher/a-54691125 (Accessed: April 25, 2023).
Dr. Edsger Dijkstra at ETH Zurich in 1994 (image by Andreas F. Borchert)
Brief introduction of graph concepts
To explain Dijkstra's algorithm, one should need to understand the basic concepts of graphs. In short, graphs are data structures that are used to denote connections between pairs of elements. We call these elements nodes. Sometimes these nodes are directed, which means for example we can go from A to B but not from B to A. As a final concept, there is the weight property which denotes the weight of a certain connection, this can be thought of as the distance between points.
Graphs are a fundamental concept in data structures. (Photo by Clint Adair on
Unsplash)
The algorithm starts at the chosen node and keeps track of the currently known shortest distance from each of the connections to the starting node and it updates when it finds a shorter path. Once it found the shortest connection that node is marked as visited and added to the path. This goes on until all the nodes have been added to the path. To demonstrate I put together an example graph to further understand the concepts.
We will keep track on what is the shortest path from node S (start) to E (end) and also list the shortest path to every individual node.
Distance: S=0, A=∞, B=∞, C=∞, J=∞, K=∞, E=∞
Unvisited nodes:{A, B, C, J, K, E}
At this point, we need to check the distance from starting node to the neighboring nodes.After that, we need to update the distance list
Distance: S=0, A=2, B=6, C=∞, J=∞, K=∞, E=∞
Then we need to select the node that is closest to the starting node and mark it as visited and also add it to our path.
Unvisited nodes:{B, C, J, K, E}
At this point, we need to further analyze the new adjacent nodes and update our distance list. Since we already analyzed node B, we only need to update the new node C.
Distance: S=0, A=2, B=6, C=7, J=∞, K=∞, E=∞
Before going through the other junction, we will backtrack and analyze the node B to C
C = 2+5 vs 6+8 shortest is 2+5=7
Distance: S=0, A=2, B=6, C=7, J=∞, K=∞, E=∞
Unvisited nodes:{J, K, E}
We continue updating our node distances
J= 2+5+15 K=2+5+10
Unvisited nodes:{J, E}
We will continue through the K junction to find the node E
node E= 2+5+10+2 vs 2+5+15+6 =19
node E becomes 19 and therefore the shortest path from S to E becomes:
S => A =>C => K =>E
Conclusion
To summarize all that we have talked about, Dijkstra's algorithm helps us find the shortest path in a given graph. It is used on navigation systems, although a modified version of some sort, because in this algorithm we check every possible node to reach our final destination, let's say we want to apply a large graph, then it becomes inefficient to check for all nodes. This algorithm is highly optimized later on to give priorities to certain nodes in order to be more efficient and faster.
Source section
- Abbas, S. H. (n.d.). What is Dijkstra's algorithm? Educative. Retrieved April 11, 2023, from https://www.educative.io/answers/what-is-dijkstras-algorithm
- Dodsworth, D. (2023, March 14). Dijkstra's shortest path algorithm explained, with examples. History. Retrieved April 11, 2023, from https://history-computer.com/dijkstras-shortest-path-algorithm-explained-with-examples/
- Navone, E. C. (2022, February 3). Dijkstra's shortest path algorithm - a detailed and visual introduction. freeCodeCamp.org. Retrieved April 11, 2023, from https://www.freecodecamp.org/news/dijkstras-shortest-path-algorithm-visual-introduction/
Photo by Андрей
Сизов on Unsplash
The role of big O notation in data structures and algorithms
To type out this notation shortly, when we want to show the runtime complexity of a linear search algorithm it is
simply: " O(n) "
This means that the time increases linearly as the number of elements increases. This deduction has been made by
considering the worst possible scenario in a given algorithm. The runtime of let's say a sorting algorithm can be as
little as one(instant), however, this is not guaranteed and therefore is not taken into account when we talk about
big O notation. Here are the different algorithms with their big O notations and their relative positions on a
linear graph.
As the graph shows, we can deduct whether an algorithm is efficient and its feasibility
before implementing it.
Combining algorithms and for loops
When combining different big O values, we must follow these mathematical rules. Firstly, let us imagine two
functions, namely f, and g which have their own runtime complexity.
If we want to combine these two algorithms and
deduct the big O notation of the combined algorithm, we would get: O(f(n) + g(n))
Consequently, when we iterate over a set of elements or arrays, we get the size of the element as n where n is the
number of elements( infinitely large) in the loop. The complexity can be expanded when we put the same size loop
inside a loop where at that point, the big O notation expands exponentially.
How code slows as data grows
To conclude this post, big O notation is an essential tool when it comes to scalability, where the feasibility of certain approaches is decided by programmers. It is certainly a great tool for anyone working with the new emerging technologies in IT, namely Artificial Intelligence and Machine Learning.
Source section
- Cormen, T., & Balkcom, D. (n.d.). Big-O notation (article) | algorithms. Khan Academy. Retrieved April 04, 2023, from https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-o-notation
- Huang, S. (2022, December 8). What is big O notation explained: Space and time complexity. freeCodeCamp.org. Retrieved April 04, 2023, from https://www.freecodecamp.org/news/big-o-notation-why-it-matters-and-why-it-doesnt-1674cfa8a23c/
- Nielsen, J. (2020). What is big O notation? jarednielsencom RSS. Retrieved April 04, 2023, from https://jarednielsen.com/big-o-notation/
Portrait of Spinoza
In his teens and early adult life, he met with plenty of people from diverse backgrounds through his time selling tropical fruit. His curiosity turned into questioning the accuracy of the Judaic religions when he started to attract attention as a heretic. This curiosity later grew when he read Pre-Adamitae (Man Before Adam) by Isaac La Peyrère. It challenged the ideas of the Catholic Church and consequently the whole Christian world by questioning the accuracy of the Bible and he supported his idea by suggesting the spread of human beings all over the world implies the nonexistence of Adam and Eve and concluded his remark by saying Bible is the history of Jews and their surrounding communities and should not be taken into account when examining the history of the world.
17th-century Amsterdam was a metropolitan heart of Europe (Photo credit: Victoria Art
Gallery)
In light of these events, he was formally excommunicated in 1656, and harsh rules were applied to anyone who dared to have any relationship with him. Despite this, his community offered help and was reluctant about his excommunication. A short time after he changed his name from Baruch to the Latin word Benedictus, both of which have the meaning of "blessed".During his time in Amsterdam, he had met with members of Collegiants and Quakers (religious groups in Amsterdam that resisted any formal creed and practice.) and joined their discord on philosophy, and formulated some parts of his philosophy from their doctrine.
"Although I have been educated from boyhood in the accepted beliefs concerning Scripture, I have felt bound in the end to embrace other views"
-Baruch Spinoza
:max_bytes(150000):strip_icc():format(webp)/QuakersBeliefs-72882874-594a9d083df78c537bac9fed.jpg)
Quakers were somewhat popular during the 17th-century
His philosophy in simple terms moves away religion from superstition, formal creeds, and practices. According to this belief only accepted scientific truths and facts were the objective truth. God is not a person who stands outside of nature; consequently, there is no one to hear our prayers to create miracle events or punish us for our sins beings aim, and most importantly, there is no afterlife. Despite all these claims he is not an atheist, quite the opposite he remained a persistent defender of God. Spinoza's God is the nature, existence, and the very fabric of the Universe and everything within the scope of imagination and therefore cannot be individuated.
Ethica today is a popular philosophical book
Humans try to understand how and why the universe works the way it does and accept the things that come from scientific pursuits rather than whine, pray, or protest the facts and events.
"Whoever loves god cannot strive that God should love him in return"
To understand, the quote suggests that only a narcissistic and egocentric person believes in God and would take an interest in bending the rules of the Universe to benefit a single person. This philosophy was influenced heavily by Stoic philosophy which argued that the protest against how things are is not a wise action but making continuous attempts to understand the world and only then bow down to the Universe. The last part of Spinoza's philosophy is, how do you understand the concept of God? Well to understand God, one must understand how the Universe and life work through natural sciences, psychology, and philosophy. One of the most critical aspects of this is it suggests we can exceed to a divine eternal perspective through studying the Universe.
As humans, we should strive to achieve scientific feats (Bill Ingalls/NASA, via Agence
France-Presse — Getty Images)
According to Spinoza, there are two ways of looking at life. We can either act egoistically from our own perspective as he called it Sub Specie Durationis (under the aspect of time) or we can see things eternally Sub Specie Aeternitatis (under the aspect of eternity). Our life as we experience with our bones and flesh may pull us towards time bound view (Sub Specie Durationis) but our reason and intelligence can give us a unique perspective it can quite literally allow us to participate in eternal royalty (Sub Specie Aeternitatis)
Why was Spinoza not popular?
The question then becomes why was Spinoza not popular for his beliefs. His famous book Ethica (Ethics) is one of the world's most beautiful books. It serves as a calming perspective regarding life. It replaces the all-powerful and sometimes angry God figure with a wise and consoling pantheistic type of God. The reason it is not popular is that religion includes far more emotion, belief, fear, and simply tradition. People stick to their religious beliefs because they like traditional values and regular events. Ethica is a well-structured book but it alone cannot contribute to people changing their beliefs. But in today's world even if Ethica is not known by a lot. Its ideas are more or less the cornerstone of western religious beliefs.
Source section
- BBC. (2009, July 3). Religions - christianity: Quakers. BBC. Retrieved March 28, 2023, from bbc.co.uk/religion/religions/christianity/subdivisions/quakers_1.shtml
- Dutton, B. (n.d.). Internet encyclopedia of philosophy. Retrieved March 28, 2023, from https://iep.utm.edu/spinoza/
- Encyclopaedia Britannica, T. E. of. (2021). Quaker. Encyclopædia Britannica. Retrieved March 28, 2023, from https://www.britannica.com/topic/Quaker
- Encyclopædia Britannica, inc. (n.d.). Dutch civilization in the Golden Age (1609–1713). Encyclopædia Britannica. Retrieved March 28, 2023, from https://www.britannica.com/place/Netherlands/Dutch-civilization-in-the-Golden-Age-1609-1713
- Hopkin, R. (2023, April 1). Benedict de Spinoza. Encyclopædia Britannica. Retrieved March 28, 2023, from https://www.britannica.com/biography/Benedict-de-Spinoza
- Nadler, S. (2020, April 16). Baruch Spinoza. Stanford Encyclopedia of Philosophy. Retrieved March 28, 2023, from https://plato.stanford.edu/entries/spinoza/
- The Dutch Golden Age - a new breakthrough for Dutch art after the ... (n.d.). Retrieved March 28, 2023, from https://artpaintingartist.org/dutch-golden-age-new-breakthrough-dutch-art-renaissance/
Firstly we will set our iterative function and set an initial real number.Our iterative function is as
follows:
What was most significant about the lunar voyage was not that man set foot on the Moon but that they set eye on the earth.
Let's say we chose 2 and the iterative values are:
1st: 6
2nd: 38
3rd: 1446
4th: 2090918
......
This value gets really big as we iterate on and on again, this means the value diverges over time. However the strange thing happens when we pick any value between -2 and 0.25 (inclusive), to show that let's say we pick 0.21 as an initial c value:
1st: 0.2541
2nd: 0.27456681
3rd: 0.2853869331535761
4th: 0.2914457016148037
5th: 0.2949405969897452
...
As you can tell by this example, this iteration is stabilizing around 0.29 and nearly never change around a significant iteration later. In mathematics this is called converging. This interval of numbers that are converging is called Mandelbrot set.
This can be shown in a simple graph:
As you can see there is nothing too interesting about that graph. However, when we add another dimension to this graph in the form of complex numbers, the graph will have the following shape where the behaviour of the iteration can be bounded which means the value will stay within a distance from the point zero but will not converge, it can converge or can diverge.
Here is the example code:
Java
public static void main(String[] args) {
double number =0.21;
double init =number;
int i=0;
while(i<8) {
i++;
number = number* number +init;
System.out.println(number);
}
}
Source section
- Mandelbrot set. from Wolfram MathWorld. (n.d.). Retrieved March 21, 2023, from https://mathworld.wolfram.com/MandelbrotSet.html
- Devaney, R. (1970, November 24). What is the Mandelbrot set? Plus Maths. Retrieved March 21, 2023, from https://plus.maths.org/content/what-mandelbrot-set
- Fractal geometry. IBM100 - Fractal Geometry. (n.d.). Retrieved March 21, 2023, from https://www.ibm.com/ibm/history/ibm100/us/en/icons/fractal/