Course:EOSC311/2025/Fractals in Nature: Mathematical Patterns in Geological Formations

There are many connections between the fields of Geology and Mathematics. This project will be highlighting the appearance and applications of fractal patterns in geological formations and processes. Fractals are a relatively new mathematical concept that highlight the geometry of irregular, non-smooth shapes. This is highly important in the field of geology, as most things in real life can only be vaguely approximated to the traditional shapes we know and are modelled poorly by these approximations as a result. With the emergence of fractal geometry however, geologists are able to identify present fractal patterns and apply the more accurate approximations to their modelling. The major presence of fractal geometry in geology is demonstrated by the numerous areas they show up. From mineral deposits and earthquake distribution to coastlines and hurricanes, fractals are integral to mathematically understanding geological formations and processes. Since the presence of fractals is varied, the applications of it touch many important parts of our lives. The mining industry finds uses in exploration and discovery of precious-metals and all sorts of hazard assessments can be evaluated by using predictive modelling made possible by fractals. The value of these applications are measured not only in money, but can even help prevent deaths in natural disasters. Due to the fact the field is relatively new, more applications are being discovered all the time with all sorts of different uses. Overall, this project aims to provide background information on fractals, where they appear in geology, and expand on its applications in the real world.

Statement of Connection and Reasoning

A Combined Major in Mathematics has many intersections with the field of geology. Mathematics allows people to quantify and compute many varied aspects in life, and geology is no outlier. Predicting the movement of tectonic plates, calculating the frequency of earthquakes, and the values for mining reservoir quantifiers - like tonnage and grade - all require math to find and understand. Many geological formations and processes are extremely chaotic with seemingly infinite complexity upon deeper and deeper magnification. Some examples include coastlines, river systems, and the distribution of earthquakes. This leads to the field of fractals in mathematics, a relatively new discipline that allows mathematicians to quantify and describe geometry of the highly irregular phenomena often found in nature. The more commonly used Euclidean geometry fails to properly describe these phenomena; as the father of fractal geometry, Benoit Mandelbrot, said: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

Mathematics is used all the time in everyday life, but as the topics become more advanced, the real-world applications become less clear. Highlighting the intersection between fractals and geology allows for an opportunity to apply abstract mathematical concepts in a practical, visual manner. Furthermore, exploring the ability to model geological processes using math expands on the relevance of mathematics to seemingly unrelated disciplines, and demonstrates how geologists apply their knowledge to the surrounding world. Many aspects of geology appear extremely chaotic, but mathematics reveals the underlying order and allows geologists to predict and adapt to it.

What is a Fractal

Figure 1: The property of self-similarity is displayed in a visual manner. It would be possible to take a snapshot from two different moments in time, and the images would look exactly the same. Source: Wikimedia Commons. (n.d.). https://commons.wikimedia.org/wiki/File:Self-Similarity-Zoom.gif

Fractals were first introduced by Benoit Mandelbrot in 1919 based on the works of many previous mathematicians. He noticed that many phenomena in nature are poorly described by Euclidean geometry, which relies on regular, smooth shapes. He hoped to remedy this problem to enable the quantization of the highly irregular, complex shapes found in reality. Thus, founding and dedicating his research to the field of fractals.

In the most basic terms, a fractal is a set or geometric curve that can be viewed at different scales and retain the same structure. This is mostly commonly demonstrated in images like the Mandelbrot Set, shown in Figure 1. The result of this property causes anything with fractal behaviour to have infinite complexity due to the repeating nature of its structure at infinitely small levels. One of the first examples published by Mandelbrot was centered around the difficulty of measuring Britain's coastline.

However, the property of self-similarity does have useful applications. It leads to a concept called scale invariance, which implies that anything applicable at one scale will be applicable at every scale. This allows mathematicians to reveal and analyze complex underlying patterns by applying their macroscopic properties and scaling it to the correct dimension. While nothing in nature is a perfect fractal, many formations in nature can be viewed approximately as a fractal. This leads to many useful applications for researchers studying phenomena with extremely complex geometry.

Fractals in Geology

Figure 2: A demonstration of the difficulty in measuring the fractal geometry of Britain's coastline. Smaller and smaller scales reveals higher levels of complexity to be measured. The fractal behaviour of a coastline causes any measurement to have some level of error. Source: Wikimedia Commons. (n.d.). https://commons.wikimedia.org/wiki/File:Britain_Fractal_Coastline.gif

Fractals have been found throughout a wide variety of fields in geology. As mentioned, formations like coastlines and rivers have fractal behaviour. Shown in Figure 2, it is easy to see how the estimated length of coastline is a function of the unit measurement length. In other words, smaller and smaller scales reveal higher levels of complexity to be measured; this is a clear example of a fractal. Even in the diagram, the scale is large enough to show the entire shape of the landmass, so it is not even close to being as accurate as it could be. However, not all examples are so trivial and can have great implications for their respective field.

One extremely important fractal pattern discovered in geology is the relationship between the magnitude of earthquakes and their spatial/temporal distribution. Earthquakes happen all across the globe and the vast majority are independent of each other. So, the behaviour is expected to be chaotic and difficult to predict. However, a global fractal dimension can be calculated implying a fractal geometry to the distribution of earthquakes around the worlds. This has been supported by the fact that earthquake magnitude and probability of occurrence have been observed to have a scale invariance. Furthermore, when considering a single region, clear patterns emerge with their own dimension that describe the probability of another earthquake occurring within a certain distance. These relationships are hoped to help correctly replicate seismicity on a macroscopic scale and improve earthquake hazard assessment.

Figure 3: An animation of a fractal mountain. Increasing the dimension and scale reveals higher levels of complexity and a more accurate image. Source: Wikipedia, the free encyclopedia. (n.d.). https://en.wikipedia.org/wiki/File_talk:Animated_fractal_mountain.gif

Another area where fractals appear is in the fractures created in rock mass due to mining. These fractures are dangerous as they can cause deadly water inrush disasters and are not clearly understood at present. However, using experimental methods, the fractal dimension of a fracture system can be calculated. A study done at Xuchang University by Binbin Yang and Yong Lui calculated the dimension of the fractures during regular mining distance intervals. Then, using digital imaging, measured the length of the fractures and formulated a relationship between fracture length and fractal dimension. Unfortunately, the fractal dimension of a fracture network varies widely under different mining stress conditions, which causes any standardization to be difficult. Still, with other parameters and more research, being able to characterize and correctly simulate mining fractures with fractal geometry is within reach and would be used to improve safety conditions in the industry.

Overall, the presence of fractals in various areas of geology highlights their significance in the field. There are countless other examples of where they appear; as large as the formation of fault lines and as small as the structure of a crystal. Though, all demonstrate the importance of understanding the fractal geometry of such phenomena. Even though challenges remain, the continued exploration of fractal dimensions holds great promise for improving models, enhancing safety protocols, and deepening our understanding of the complex systems on our world.

Applications of Fractals in Geology

Identifying fractals in nature is only the beginning of the intersection between geology and mathematics. After identification, researchers are discovering many varied applications for this behaviour. It allows geologists to use mathematical modelling to predict occurrences of formations, accurately simulate processes on different scales, and formulate relational laws between variables.

One real world application for fractals in geology lies in the mining industry. An important part of the development stage in the mining cycle is a description of the spatial distribution of mineral deposits. Research has shown that fractal dimensions are accurate in describing naturally occurring statistics for these deposits that provide useful implications for exploration. Furthermore, the observed fractal dimension can be compared to other known values to determine the viability of exploration. For example, it has been used to describe the clustering of hydrothermal precious-metal deposits in western United States. The distributions behave on two scales that cover a radial distance of up to 1000km away from the vent. The fractal geometry describe a radial-density law, where the deposits per square kilometer can be accurately calculated. Another example is in the Baneh-Saqqez zone, northwest of Iran. It is shown that the gold mineralization has comparatively high fractal dimension, which allows geologists to relate it to regional structural features and facilitates the discovery of new deposits. Overall, the emerging applications fractals in the mining industry will enable higher precision exploration and a dimensional measure to evaluate viability.

Many natural disasters on Earth, like volcanoes and earthquakes, are the result of geological processes below the surface of the crust. Others, like hurricanes, take place in highly chaotic systems. Due to these factors, predicting the size, location, and timing of these events is extremely difficult. In the past, statistical methods were used to make these forecasts. However, when using fractal methodology, a level of information emerged that had never before been seen in the old methods. Previously, fractals have been discussed to be patterns that repeat in space. However, this repeating nature can also be present when using time as the dimension. One example of this is demonstrated by researcher Christopher Barton from the United States Geological Survey. He compared fractal formulas for a hurricane's size and frequency of wind speed to historic records for previous hurricane's landfall location and timing. He then found that it was possible to predict the wind speed of a hurricane at the moment of landfall on a given location. This forecast information is invaluable, as government relief and emergency agencies can use it to enable focused efforts in at-risk regions. The analysis of fractal patterns found over the last 100 years which helps geologists predict the probability of future, potentially deadly events demonstrates a highly important application of fractals.

These applications highlight the significance of researching the presence of fractals in geology. Not only can it help improve processes in the mining industry, but it is extremely useful in areas like predictive modelling. As discussed, precise forecast information is highly important when it comes to hazard assessment, and the fractal dimension excels in this area. These applications barely scrape the surface of the benefits that could be found, but their importance is already clear. Furthermore, due to the fact the concept of fractals is relatively new, there is a strong indication that many more useful applications will still be found. For this reason, it is highly important that research continues on the application of fractals in geology.

Conclusion / Your Evaluation of the Connections

In conclusion, understanding the underlying fractal geometry in geological phenomena is a highly important intersection between the fields of mathematics and geology. Fractals are uniquely qualified to address the highly irregular and complex geometry that is commonly found in geological formations and processes; processes that Euclidean geometry fails to properly describe.

Throughout this project, the repeated and varied presence of fractals across many areas of geometry were highlighted to expand on the real-world examples where they appear. This demonstrate that fractals are not just an abstract mathematical concept, but has practical implications that geologists are able to harness to make sense of a seemingly chaotic system. From global scales measuring earthquake distribution to the macroscopic scales describing mineral deposit distribution, and finally, to the microscopic scale of rock fractures and structures, fractals provides a means of modelling and predicting geological behavior across many scales.

With the significant presence of fractals in geology, the applications for it have far-reaching effects. Applications in the mining industry are invaluable, as fractal dimensions can be used in resource exploration, discovery, and evaluation of site viability. These factors will all serve to improve the cost-efficiency for mining companies searching for new mineral deposits. Furthermore, applications in nature disaster research has value in other ways. Improving disaster precision, fractals enable precise hazard assessment and early-warning predictions. In all, this information is used to help offset disaster severity, which ultimately saves lives and reduces property damage. These contributions are significant, and made even more prevalent by the fact that they arise from a relatively new mathematical field. This indicates a strong potential for future discoveries and innovations.

Overall, fractals serve as the tangible bridge between geology and mathematics. There are many highly complex systems in geology and processes happen on scales that are difficult to study. Fractals have opened a new avenue of research in geology that enhances our understanding of these phenomena greatly. As the field develops, it is likely that many more applications will emerge, which will certainly cement the relationship between these fields as deeply valuable.