Biophilic Architecture and its Impact on Cognitive Function

What is biophilic design? Imagine stepping into a classroom, free from artificial lighting, plastic chairs, and blank walls— replaced instead with natural light, wooden furnishing, and flourishing greenery. This is the essence of biophilic design. Biophilic design is a modernized approach to architecture that is rooted in the connection between humans and nature. Some of the style’s key characteristics include biomorphic or organic shapes and patterns, natural and open lighting, incorporation of greenery, and the use of natural materials. In description and in image, biophilic architecture is a harmonious blend of indoor and outdoor environments. Beyond the surface, however, this design philosophy nurtures psychological health and well-being. 

Biophilic design emerged from a philosophical theory called biophilia. Quite literally meaning the love of life, biophilia is the term coined by psychoanalyst Erich Fromm in 1973 to define the exploration of the essence of humanity. Fromm’s belief was that mortality salience and the awareness of existence disrupt the connection humans share with nature, ultimately leading to turmoil and anxiety. Biophilia, therefore, would be the reunion of humanity with their natural tendencies to love nature, returning a sense of well-being. Sociobiologist E.O. Wilson expanded upon the definition of biophilia as an aspect of human evolutionary survival and a “deep and complicated process in mental development” (Söderlund and Newman, 2015). From these individuals, modern scholars have concluded that incorporating biophilia in urban environments is essential for cognitive function, stress reduction, and ubiquitous fulfillment (Söderlund and Newman, 2015).  Through this idea, biophilic design and biophilic architecture were born.

While there are multiple principles that are core to the architectural style, the three integral ones are natural lighting, natural materials and shapes, and greenery. One method of incorporating natural lighting into biophilic spaces has been through the use of low emissivity (Low-E) glass to regulate heat transfer, glare, and ultraviolet (UV) damage in buildings. This happens through applying a thin, metallic, and mostly silver coating in between the layers of the glass. Silver is considered ideal due to its outstanding electrical conductivity, making it the standard against which all other metals are compared. It is spread so thin that an individual layer of a Low-E coating is 500 times thinner than a strand of hair or a piece of paper. These coatings help reflect infrared (IR) and UV rays while allowing visible light to pass through (Guardian Glass, n.d.). Beyond the aesthetics, there are significant benefits of natural lighting. In 1999, the Heschong Mahone Group analyzed the effects of natural daylight in three school districts. Its results showed a 20% to 26% uptick in learning speed, a 5% to 14% improvement in test scores, and a reduced absenteeism rate of three and a half fewer days per year for students in schools with natural light. (Heschong, 1999). 

Biophilic design’s extensive use of natural materials in office environments contributes positively to cognitive function. Because of this, biophilic architects lean toward using wood foundations over steel or concrete. In order for their projects to produce the same high-quality results as the other materials, architects will typically plan to use cross-laminated timber (CLT) or glued laminated timber (Glulam). CLT is a type of wood product that is made of kiln-dried dimensional lumber to ensure that it is straight and split-free. The lumber pieces are then perpendicularly glued together with adhesive to create sturdy and durable panels. Glulam is similar, however the lumbers are placed parallel to each other. This means that Glulam is better used for vertical supports such as columns or beams, whereas CLT is better used for walls, floors, and ceilings (Kallesoem Machinery, n.d.). Like natural lighting, the exposure to wood as a natural material can contribute to stress reduction through its organic patterns that reflect nature. In 2019, researchers Michael David Burnard and Andreja Kutnar conducted a study on how different office environments influence stress levels in individuals. Participants completed a written test, watched a stress-inducing video, and proofread their work, all while seated at either oak or white plastic tables. During the study, researchers measured heart rates and cortisol levels— both indicators of stress—to analyze how these environments might impact stress responses. Saliva samples collected throughout and heart rate monitors revealed that participants seated at oak tables had generally lower cortisol levels and heart rates than those at the plastic white tables (Burnard and Kutna, 2019). These findings suggest that exposure to natural materials helps create calming environments that contribute to reduced stress.

One method of incorporating greenery into modern biophilic design is through living walls. Living walls are vertical structures covered by vegetation and are sustained through three types of systems: the panel system, the felt system, and the container system. All systems go according to name; plants grow on panels in the panel system, on fabrics in the felt system, and in containers on a trellis in the container system (Jiang, 2022). The multitude of systems is due to the different environments in which living walls are installed. For example, indoor versus outdoor or hot versus cold environments. Exposure to greenery is one of the most straightforward methods of improving one’s well-being. A significant number of studies have been conducted on the benefits of being in proximity to plants, showing that having plants in work, living, and school environments holds a direct correlation to improving attention span, mental clarity, and task performance (TEDx Talks, 2022).

Biophilic architecture is not just a building style but a lifestyle as well. Combating the design of urbanism and its stressors to paint an aesthetic and healthier future shows how society has increasingly prioritized mental health and well-being, furthermore upholding the wellness of the planet. This indicates the need for changes in office and education spaces in order to ensure they uphold individual well-being. While it may seem insignificant, installing a couple of plants, adding wooden furnishing, or even just opening a window might make all the difference.

References

Burnard, M. D. (2019, September 4). Human stress responses in office-like environments with 

wood furniture. Taylor & Francis. Retrieved December 18, 2024, from https://www.tandfonline.com/doi/full/10.1080/09613218.2019.1660609#d1e171 

Cross-laminated timber (CLT). (n.d.). naturally:wood. Retrieved December 18, 2024, from 

https://www.naturallywood.com/products/cross-laminated-timber-clt/#:~:text=What%20is%20CLT%20used%20for,insulated%20wall%20and%20roof%20sections 

Hall, C., & Knuth, M. (2019, March 1). An Update of the Literature Supporting the Well-Being 

Benefits of Plants: A Review of the Emotional and Mental Health Benefits of Plants. Journal of Environmental Horticulture. Retrieved December 18, 2024, from https://meridian.allenpress.com/jeh/article/37/1/30/430948/An-Update-of-the-Literature-Supporting-the-Well 

Heath, O. (2022, March 1). Biophilic Design: Supporting People and Planet | Oliver Heath | 

TEDxUCL [Video]. YouTube. https://www.youtube.com/watch?v=wbb5uzj1b3I 

Jiang, S. (2022, December 8). The Approachable Green: Integrating Living Walls in Biophilic 

Interior Spaces. The Field. Retrieved December 18, 2024, from https://thefield.asla.org/2022/12/08/the-approachable-green-integrating-living-walls-in-biophilic-interior-spaces/ 

Low-E glass. (n.d.). Guardian Glass. Retrieved December 18, 2024, from 

https://www.guardianglass.com/me/en/our-glass/glass-types/low-e-glass

Olt, M. (2021, January 9). Biophilia, the future of architecture | Matthias Olt | TEDxSeattle 

[Video]. YouTube. https://www.youtube.com/watch?v=OXZzsty5BGY 

Söderlund, J., & Newman, P. (2015, December 10). Biophilic architecture: a review of the 

rationale and outcomes. AIMS Press. Retrieved December 18, 2024, from https://www.aimspress.com/article/10.3934/environsci.2015.4.950 

What is the difference between CLT and glulam? (n.d.). Kallesoe Machinery. Retrieved 

December 18, 2024, from https://kallesoemachinery.com/the-green-products-of-the-future/what-is-the-difference-between-clt-and-glulam/ 

Anthrobots: Robots from Cells

What is a robot? When most people hear about robotics and Artificial Intelligence, they probably picture humanoid robots like the ones that Boston Dynamics or Tesla are making. According to the Oxford Dictionary, robot means “a machine resembling a human being and able to replicate certain human movements and functions automatically” or “a machine capable of carrying out a complex series of actions automatically, especially one programmable by a computer” (“Robot, N.2 Meanings, Etymology and More | Oxford English Dictionary,” 2024). While both of these definitions apply to the humanoid robots that large tech companies are making, they can include other types of robots also. For example, some robots do not look remotely like a human except for something that resembles an arm. Some do not  even have that. One example of a robot like this is a biological robot. A biological robot is a robot that is based on an animal. Some examples for this are Spot from Boston Dynamics or one of many snake looking robots that are being developed. While not all biological robots fall under the category of soft robot,  a robot made from pliable material like elastomers or biological tissue, many do. Both of these materials allow a robot to move around in ways that metal would not. Some biological robots mimic animals such as snakes or eels, while others are actually their own organism. One subset of a biological robot is a living robot, a robot that is made from cells and operates on its own. The two types of living robots that have been made are Xenobots and Anthrobots with the primary difference being the source material for the robots. Xenobots are named for the species of frog they were taken from, Xenopus Laevis, while Anthrobot comes from the Greek root Anthro meaning man.

Xenobots are a type of living robot created from frog skin and heart cells derived from frog embryos (Brown, 2020). Unlike traditional robots made from metal or plastic, these robots are composed of organic material. Scientists from Tufts University, the University of Vermont, and Harvard University observed that Xenobots naturally moved in circles and pushed pellets into piles. To optimize their design for this behavior, researchers used a supercomputer to model hundreds of different configurations. They eventually settled on a Pac-Man-like design. The cells were then assembled into this shape using tiny forceps and an electrode to bond them (Brown, 2020).

Using the pacman shape and natural tendency to make piles, the scientists at UVM were able to get the Xenobots to reproduce in a way that is not seen in nature. Instead of splitting through mitosis or creating gametes, Xenobots push other cells together into a new organism. Before the pacman shape, whenever the Xenobots replicated themselves, after only one to two generations it looked like a ball and did not function very well (Kriegman et al., 2021). Scientists believe that the kinematic self-replication that the Xenobots do, could be proof of abiogenesis. Abiogenesis is the belief that life on earth formed from non-living materials and that it has gotten progressively more complex over time. Beyond being a proof of concept for living robots and kinematic self replication, this type of robot can be used for an incredible array of medical procedures including delivering medicine and targeting cancer and blood clots.

This thinking is what led to the creation of the Anthrobot, a living robot made out of human cells rather than frog ones. Anthrobots can also form themselves in a petri dish, rather than needing humans with tweezers to assemble them. These Anthrobots are made out of human lung and tracheal cells because the cells have cilia along their border. In the human body these cilia are used to clear the throat whereas in the robots they are used for mobility. Like the Xenobots, Anthrobots possess the ability to heal themselves. This ability gives scientists an easy way to study how the human body repairs itself. Additionally, having the ability to quickly make anthrobots out of people would be helpful for creating noninvasive robots to clear arteries or distribute medicine (Silver, 2023).

Adding on to this discovery, further also found that cells can still be made into a Xenobot or Anthrobot despite an organism's recent death. They discovered that the cells could be used for about six weeks before they stopped working. Scientists are not fully sure why this happens, but the leading idea is that cells are very similar to electrical circuits due to the channels and pumps in the cell membrane. Then due to decomposition, after six weeks the channels and pumps do not line up right or are gone completely. Understanding why the cells can still be used, and subsequently why they later cannot be used could be used to engineer robots that deliver medicine and then decompose themselves (Orf, 2024).

While there have been many  advancements in the biorobotic field, many more are still being made. As of right now there is not an ethical issue Xenobots and Anthrobots because they lack a nervous system. However, with the jumps that this science is making, it could end up being an ethical dilemma. Currently, despite major advancements, they are still only in labs, however in the coming years they may be used in all the fields of science and could become everyday tools for the medical field.

References

Brown, J. E. (2020, January 13). Team builds the first living robots. UVM.

     Retrieved October 31, 2024, from https://www.uvm.edu/news/story/

     Team-builds-first-living-robots

Cell Death. Popular Mechanics. Retrieved November 13, 2024, from https://www.popularmechanics.com/science/health/a62244774/biobots-third-state/

Gumuskaya, G., Srivastava, P., Cooper, B. G., Lesser, H., Semegran, B., Garnier,

     S., & Levin, M. (2023). Motile living biobots self‐construct from adult

     human somatic progenitor seed cells. Advanced Science, 11(4).

     https://doi.org/10.1002/advs.202303575

Kriegman, S., Blackiston, D., Levin, M., & Bongard, J. (2021). Kinematic

     self-replication in reconfigurable organisms. Proceedings of the National

     Academy of Sciences, 118(49). https://doi.org/10.1073/pnas.2112672118 

Orf, D. (2024, September 20). Scientists Have Uncovered a 3rd State of Life, Which Starts After 

Robot, n.2 meanings, etymology and more | Oxford English Dictionary. (2024). Oed.com. https://doi.org/10.1093//OED//4330523394

Silver, M. (2023, November 30). Scientists build tiny biological robots from

     human cells. Wyss Institute. https://wyss.harvard.edu/news/

     scientists-build-tiny-biological-robots-from-human-cells/

Mathematics in Finance: The Black-Scholes-Merton Model

Did you know that approximately 20% of math graduates work in financial institutions? Quantitative analysis is an area of finance that addresses financial decision-making and strategies through applied mathematics and statistical modeling. It is used by banks, insurance companies, and hedge funds in various ways, with the primary goal of making optimal investments, loans, and insurance plans. Quantitative systems are used to determine the price a company values its products at (Kenton, 2023), the amount of aid a family in need should receive, and the loans that are given to local businesses (Supplemental Nutrition Assistance Program (SNAP) | Food and Nutrition Service, 2024). Because of these systems' wide effects on the economy, it is important to understand the basics of them.

The Black-Scholes-Merton (BSM) model is a prominent example of quantitative finance. The equation was the focus of a prominent 1973 research paper written by Fischer Black & Myron Scholes, and edited by Robert Merton. It is used to estimate the value of option contracts. An option contract grants an owner the right to buy a stock (call option), or sell a stock (put option) on a certain data, at a set price. For example, let’s hypothesize an option contract for the stock of the company Apple (AAPL). The stock is currently at $100. One could buy a contract which allows them to buy the stock at $100, for the next 90 days. If they owned an option contract for the company, and the stock went up to $200, they could exercise their contract, and buy the share for $100. Option contracts generally trade above current stock price. It is important to note that the BSM model only works on option contracts which can exclusively be exercised on their expiration date. There is a key difference between European and American option contracts, in that American option contracts can be executed any time within a certain period, while European contracts can only be executed on a certain day, after a certain period. Therefore the BSM model only properly works for European contracts.

The model relies on a few key assumptions. Firstly, stock prices have a log-normal distribution, meaning that their price cannot fall below zero and that the returns on the stock are normally distributed, allowing for predictable changes in asset price over time. This assumption is based on the concept that while stocks can experience significant changes, they are more likely to show gradual growth over time (Chen, 2022). Secondly, that stock prices follow a random walk, with constant volatility. In simpler terms, random walk refers to the assumption of the model that stock prices have an equally random chance of increasing or decreasing, and random volatility refers to the assumption that these changes occur to a constant degree. The model then requires the variables pictured above. A strike price is the price at which the option can be bought on the expiration date. This is typically near the price of the stock on the date where the option contract is bought. The risk-free interest rate is the theoretical rate of return an investor could make without risk (this is typically the current rate of US bonds, which are extremely low risk). The model itself can be broken into three parts:

• D1: Calculates the theoretical price of the option 

• D2: Utilizing D1, calculates the probability that the option will be exercised at expiration, considering volatility, and the time remaining until expiration

• C: Final theoretical value of option*

*It’s important to note that N(D1) and N(D2) represent the cumulative probability of a normal variable (a variable that follows the standard normal distribution) being less than or equal to D

While the BSM is a very useful tool for brokers, banks, and other financial institutions, it is important to understand its drawbacks. First, it assumes constant volatility. In reality, volatility can vary dramatically depending on market conditions and investor sentiment. The basic BSM model also does not account for transaction costs, dividends, or taxes. Because of this, more expansive versions have been developed, and it is often used in conjunction with other tools. And lastly, as previously mentioned the BSM model only works for European options. Despite these disadvantages, the BSM remains a powerful tool for investing. Gaining an understanding of its workings can provide deeper insights into financial modeling and market dynamics (Hayes, 2024).

References

Chen, J. (2022). European Option: Definition, Types, Versus American Options. Investopedia. https://www.investopedia.com/terms/e/europeanoption.asp

Hayes, A. (2024). Black-Scholes Model: What It Is, How It Works, and Options Formula. Investopedia. https://www.investopedia.com/terms/b/blackscholes.asp#toc-how-the-black-scholes-model-workKenton, W. (2021). Log-Normal Distribution: Definition, Uses, and How To Calculate. Investopedia. https://www.investopedia.com/terms/l/log-normal-distribution.asp

Kenton, Will. (2023) Quantitative Analysis (QA): What It Is and How It's Used in Finance. In Investopedia

Supplemental Nutrition Assistance Program (SNAP) | Food and Nutrition Service. (2022). Usda.gov. https://www.fns.usda.gov/snap/supplemental-nutrition-assistance-program

Fractals: Mapping the Patterns of Chaos

How can we predict the unpredictable? Fractals hold the key to harnessing the chaos of our universe. Fractals are beautifully mysterious shapes. They walk the line that separates chaos and order and exist between the three dimensions we experience. These geometric shapes have repeating patterns, meaning that the shape of the fractal is repeated infinitely throughout itself. This creates captivating models such as the Mandelbrot set and Sierpinki’s triangle (Britannica, 2024). But fractals are not just pretty shapes. They exist everywhere in nature, from snowflakes to the layout of dark matter in the universe, and understanding them can help scientists predict the unpredictable. 

Mathematically, fractals are described as geometric shapes that can be infinitely broken into pieces, each of which is a scaled version of the original. This property is called self-similarity. Additionally, their dimensions are fractional. Humans exist in three dimensions, the shapes analyzed in geometry are two-dimensional, and the lines represented on graphs are one-dimensional. Fractals however, are none of these. Their dimensions are not whole numbers but rather are between whole numbers (Fractal Foundation, 2018). But perhaps the most enticing property of mathematical fractals is their ability to model chaos. Chaos in math is a sequence where a slight change in initial position results in a completely different final position (Britannica, 2024). As an example, take two adjacent air particles. While they start out only a few nanometers apart, after minutes on a windy day they can end up miles apart (Britannica, 2024). Understanding chaos allows scientists to predict chaotic systems like the stock market, weather, or star clusters (Britannica, 2024). Most fractals have a connection to a chaotic system and are thus very intriguing to scientists (Fractal Foundation, 2018). 

For an example of a fractal, draw an equilateral triangle. Then, draw another one with vertices at the midpoints of the first one. This should create four congruent triangles. Repeat this process as many times as possible with the three outermost triangles, leaving the middle one empty. The resulting shape should look like this:

This is Sierpinki’s triangle, named after the mathematician Wacław Sierpiński who discovered it. It is a straightforward and comprehensive example of a fractal. Each part of the larger triangle is a scaled-down version of the original.

Sierpinski's triangle can also be derived through a chaotic system. The vertices of a triangle are drawn and a random point p, chosen inside of that shape, is also marked. Then, one of the vertices is selected randomly, and a new point, q, is drawn halfway between p and the vertex. The process is repeated with point q. The resulting shape is Sierpinski’s triangle. 

Another famous fractal that models chaos is the Mandelbrot set. The Mandelbrot set is famous for being an aesthetic masterpiece despite its humble mathematical formula. The formula is defined as Zn+1 = Zn2 + C where Z is a set of numbers, Zn is the nth term in the sequence, Zn+1 is the following number in that sequence, and C is a complex number. The way the graph is created is that Z1 (the first number in the sequence) starts as 0, and C is any point on the complex plane (the plane that contains real and complex numbers (a+bi) rather than (x,y) coordinates). The output Z2 (the second number in the sequence) is determined by adding Z12 and C. Then, Z3 is found by squaring Z2 and adding C, and so on. This will result in one of two tendencies: either Zn will expand outwards approaching infinity or it will collapse and approach 0. By graphing only the complex numbers C which, when put in this equation, approach 0, the Mandelbrot set (named after Benoit Mandelbrot, its creator) is defined (Fractal Foundation, 2018). This shape is a graph of a chaotic system since small differences in the C value can result in completely different outcomes. While the Mandelbrot set and its many variations generally remain only for mathematical analysis, there are many fractals that have practical applications.

Fractals can be found nearly everywhere in nature and arise in all different areas of science. In fact, many things in our everyday lives – trees, mountains, the circulatory system, snowflakes, leaves, rivers – are fractalesque (Montana State University, 2011). While nothing in nature can be a true fractal (infiniteness, going against the fundamental principles of nature, requires that it remains confined to the mathematical realm), we come across many systems that are fractal-like, meaning that they are self-similar, just not infinitely so. Here are three principles of science that have their roots in fractals:

Chemistry

Snowflakes are great examples of fractals and a chaotic system. Snowflakes form in layers which are created all throughout their journey to the ground. Each layer forms according to factors in its environment such as temperature, humidity, and speed. As the snowflake falls, it passes through different environments, driving each layer of the arms to look different from the previous one, yet the same as the other arms on that layer. At the end of its journey, the snowflake has picked up fractal-like properties. Each arm is similar to the others, and each branch of the arms are similar to the arms themselves, and so on. These fractals also model a chaotic system. Two adjacent raindrops, on their path to the ground, will experience vastly different environments and usually reach the ground far apart. This is the reason why each snowflake is unique (NOAA, 2022). 

Physics

Lichtenberg figures occur when high voltage charges pass through an insulator (a non-conductive material). The electricity spreads out through the insulator in a lighting-like shape. These figures, first discovered by German scientist Georg Lichtenberg, often appear as scars on lightning strike victims since the human body acts as an insulator and lighting is a natural occurrence of very high voltage (Stone Ridge Engineering, 2024). The figures look like this:

The high voltage forces electrons into the insulator. Once they enter, wanting to minimize the potential energy of the electrons, they spread out to cover the maximum area in order to decrease the voltage drop. The shape they make is the Lichtenberg figure. These can also model chaos since any given electron has hundreds or thousands of different paths it can take as it spreads through the insulator and any slight difference in the initial position of an electron can significantly change its path (Stone Ridge Engineering, 2024). 

Biology

The body is a masterpiece of elegant functionality. The paradigmatic example of this is the circulatory system. The job of the circulatory system is to provide oxygenated blood to the cells. To bring blood to each of the 36 trillion cells in the human body, the circulatory system utilizes a fractal branching structure, similar to that of the Lichtenberg figure. The main blood vessels, the ones that directly connect to the heart, split off, creating smaller vessels that in turn split into even smaller ones. This self-similar pattern creates a network of blood vessels that reach the farthest corners of the body. In fact, as a result of the fractal-shaped network, there is thought to be about 60,000 miles of blood vessels in the human body, which is enough to circle the Earth twice (Fractal Foundation, 2018).This feat of evolution is designed to maximize the reach of the blood while minimizing the number of vessels required (Montana State University, 2011).

Observing fractals helps scientists understand the world. In a system that seems chaotic and unpredictable, fractals can be used to predict them. This allows scientists to learn about complex ideas like brain waves and bacterial growth, and develop new technologies that depend on fractals, like cancer detection and antennas (University of Waterloo, 2001) (Obert, et al.,  1990)(Montana State University, 2011). While Normal cells usually reproduce in a fractal pattern, cancerous cells are much less organized. To find abnormalities, scientists measured the dimensions of the growth pattern of healthy cells and cells at different stages of cancer. By comparing these dimensions to the ones detected in the patient, they were able to isolate cancerous regions (Elkington, et al., 2022). This has major implications for the future of cancer detection and treatment. 

In the field of bacteriology, scientists also found fractal structures. They discovered that some bacteria grow in fractal patterns also with fractional dimensions (Obert, et al.,  1990). This is important because it can help biologists to predict how bacteria will grow which could disrupt many biological industries. For the technology sector, fractals can help build antennas. These are not the antennas that stick up from old TVs, but rather the small electronic ones that communicate with other devices through frequencies such as the ones in your phone or wireless earbuds. These are very useful but they have their limitations. Most antennas can only receive frequencies within a small range. To expand the range of antennas, fractals can be used. Fractals are self-similar on different scales. This means that each iteration can detect a different frequency. With only a few iterations of fractal geometry in their construction, the upper and lower limits of the antenna can be stretched significantly (Yale, n.d.). These are just a few ways that demonstrate how fractals have and will continue to have fascinating impacts on our life and the future of science. These beautiful and mysterious figures deserve more recognition. They are the key to discovering the order in the chaos that is our universe.

References

Britannica, T. Editors of Encyclopaedia (2024, October 25). fractal. Encyclopedia Britannica. https://www.britannica.com/science/fractal

Britannica, T. Editors of Encyclopaedia (2024, October 25). chaos theory. Encyclopedia Britannica. https://www.britannica.com/science/chaos-theory

Elkington, L., Adhikari, P., & Pradhan, P. (2022, January 7). Fractal Dimension Analysis to Detect the Progress of Cancer Using Transmission Optical Microscopy. MDPI. Retrieved November 13, 2024, from https://www.mdpi.com/2673-4125/2/1/5

Fractal Foundation. (2018). Fractal Blood Vessels. Fractal Foundation. Retrieved November 6, 2024, from https://fractalfoundation.org/OFC/OFC-1-3.html

Fractal Foundation. (2018). Mandelbrot Magic. Fractal Foundation. Retrieved November 5, 2024, from https://fractalfoundation.org/OFC/OFC-5-5.html

Fractal Foundation. (2018). What are Fractals. Fractal Foundation. Retrieved November 5, 2024, from https://fractalfoundation.org/resources/what-are-fractals/

Montana State University. (2011, October 13). Beautiful math of fractals. Phys.org. Retrieved November 5, 2024, from https://phys.org/news/2011-10-beautiful-math-fractals.html

NOAA. (2022, December 21). How do snowflakes form? Get the science behind snow. noaa.gov. Retrieved November 5, 2024, from https://www.noaa.gov/stories/how-do-snowflakes-form-science-behind-snow

Obert, M., Pfeifer, P., & Sernetz, M. (1990). Microbial growth patterns described by fractal geometry. Journal of bacteriology, 172(3), 1180–1185. https://doi.org/10.1128/jb.172.3.1180-1185.1990

Stone Ridge Engineering. (2024, June 4). What are Lichtenberg figures, and how do we make them? Captured Lightning. Retrieved November 5, 2024, from https://capturedlightning.com/frames/lichtenbergs.html#What

University of Waterloo. (2001, October 1). Top 5 applications of fractals. University of Waterloo. Retrieved November 6, 2024, from https://uwaterloo.ca/math/news/top-5-applications-fractals

Yale. (n.d.). Fractal Antennas. Yale Math. Retrieved November 13, 2024, from https://gauss.math.yale.edu/fractals/Panorama/ManuFractals/FractalAntennas/FractalAntennas.html

How Wifi Works

The way information gets to a user seems magical: how does a computer readily have access to information without some sort of cable feeding it all the information it needs to know? Also, how does it do it so quickly? Wi-Fi, which stands for Wireless Fidelity, allows people to readily access the Internet without having to be connected to a cable, making the process of communication and getting information much easier for everyone across the globe.

First, when a user requests information from the computer, the computer’s Wi-Fi chip converts all the requested information into a lot of binary digits, or bits. These bits are the 0s and 1s that all computers work with on the lowest level of their operation. Once this is done, the binary information is translated into radio waves that is then sent to a router. Most of these radio waves are either at a frequency of 2.4 gigahertz or 5 gigahertz (2.4 or 5 billion waves per second). The router then converts the frequency of the radio waves back into the binary information that the user requested, and sends that information to the Internet to retrieve the information the user asked for through an Internet cable. Once this process is complete and the information is retrieved, the same thing happens in reverse to send the information back to the computer. This process happens so quickly that most routers can handle 54 million bits of information per second! Wi-Fi, contrary to popular belief, is not magic, rather amazing technology that allows millions of pieces of information to register on a user's screen in a single second, giving comfortable and accessible information to all those who desire it.

Supermaneuverability

In modern air combat, science and engineering, coupled with new technologies have strongly influenced the capabilities of fighter aircraft. These aircraft are faster, deadlier, more advanced, and more maneuverable than ever before. However, one intriguing concept has revolutionized dogfighting and modern air combat. The concept of supermaneuverability. 

Supermaneuverability is defined by the United States Defence Technical Information Center (DTIC) as “the capability of a fighter aircraft to execute tactical maneuvers with controlled side slipping and at angles of attack beyond maximum lift.” These aircraft can “maintain a high degree of maneuverability below corner velocity, and at least limited attitude control without altitude loss below stall speed. Such an aircraft is capable of maneuvers that are impossible with a purely aerodynamic design.” Explained simply, supermaneuverable aircraft can conduct maneuvers at extremely high angles of attack (the angle at which the chord of an aircraft's wing meets the relative wind), even when at stall speeds, where normal aircraft would experience a loss of lift and would depart flight. 

In order to be supermaneuverable, an aircraft must be designed differently. One of these unique characteristics that supermaneuverable aircraft feature is thrust vectoring. According to NASA, “Some modern fighter aircraft can change the angle of the thrust by using a movable nozzle. The ability to change the angle of the thrust is called thrust vectoring, or vectored 

thrust.” As shown by this free-body diagram from NASA, aircraft with thrust vectoring capabilities can change the angle of their thrust, thus changing the direction of the net force. Supermaneuverable aircraft can use thrust vectoring technology to maintain attitude control and also increase their maneuverability by being able to direct their thrust according to the direction of their maneuvers. Because of these unique capabilities, many of the world’s most powerful dogfighting aircraft, that is aircraft that excel in within-visual range combat against other aircraft, are supermaneuverable. This includes the United States Air Force’s Lockheed Martin F-22 Raptor and Russia’s Sukhoi Su-57 among others. All in all, supermaneuverability allows dogfighting aircraft to be more maneuverable, controllable, and deadlier than ever before. 

The Butterfly Effect

Imagine a butterfly flapping its wings in the Amazon Rainforest. It might seem like an incredibly trivial action with zero significance, however, according to the butterfly effect, the inconsequential flap of its wings can set off a chain reaction that could change the world forever. 

This phenomenon was first discovered by Edward Lorenz in the 1960s and explains that small actions can have large consequences because systems are extremely dependent on initial conditions. Lorenz was a professor of meteorology at MIT and wanted to use new computer technology to be able to more accurately predict weather cycles. To simulate a physical system such as the weather, one must solve differential equations, which predict how a system changes over time. Lorenz found that when he rounded one variable from .506127 to .506 the results changed dramatically. This created the idea of “sensitive dependence on initial conditions.” The understanding of the tiniest of changes leading to wildly different outcomes. This discovery led to a new branch of mathematics known as chaos theory that studies the behavior of systems that are “unpredictable” due to their extreme sensitivity to initial conditions.

Chaos Theory and the Butterfly effect have many real world applications including the flow of liquid, weather, population cycles, stock market rates and so much more because of how many variables exist in the world we live in and how unpredictable it is. The butterfly effect highlights just how interconnected the world is and the inherent unpredictability of it.