032 033 This randomness is what is necessary for all cryptographic keys and a random occurrence which cannot be corrupted by cyber technology, so is the exact solution we need to improve data protection in the 21st century. You may be asking how you could base an entire security system on lights in the sky that everyone can see. This is a valid question; however, the first layer of lights that you see is not the only part of the polar lights that is used. Instead, the aurora borealis extends for hundreds of meters into the atmosphere with many distinct layers with different colours and emission spectrums. Accurately taking data from the aurora borealis would be nearly impossible from ground level even with the most advanced imaging technology. In order to take data from the aurora borealis drones would need to be implemented which can be flown level with the layers of lights to read them. As drones are the only way to actively take data from the aurora borealis, the airspace could be declared a no-fly zone in the interest of protecting the data. The process of creating encryption keys would need a lot of complex coding behind it as the only emissions from the aurora borealis are just different frequencies of light. An example of this coding could be that each colour would stand for a specific key and the combination of colours in each layer of the aurora borealis could represent an individual bit of encryption and the image rendered from the drones could be used to generate a random key. Feasibility While the idea presents many positive traits and solves the main problem of online security, there are some concerns about the scale of the operation. A major limiting factor is the ability to accurately track and retrieve data from the aurora borealis. Each layer of the lights would need a drone to get up close and read. A direct result of this would be the large amount of funding and preparation needed to carry out such a method. The organisation which runs this security system would also need to invest in a security system to protect the airspace. This could even require military help due to the expansive nature of the aurora borealis. Extreme weather conditions could also be another limiting factor as drone technology relies heavily on the weather being suitable for flights. Blizzards and storms, which are common in polar regions, would heavily impact the sustainability of the operation resulting in no more new keys being created. Conclusion In conclusion, with the ever-increasing threat to cybersecurity, it is imperative to explore innovative and unconventional solutions. The aurora borealis provides a much- needed incorruptible cyber key from a highly complex system. The development of this key would make data security in the 21st century much stronger and able to withstand the advances in technology for the foreseeable future. The aurora borealis is not just a beautiful spectacle, but it may hold the key to securing our digital future. Gabriel’s Horn Sankarshan Bhattacharyya Gabriel’s horn (also known as Torricelli’s trumpet) is a hypothetical solid of revolution that ostensibly has infinite surface area but finite volume. It is modelled by the function y=1/x (1≤x<∞), which is rotated 360° or 2∩ radians around the x-axis. The special mathematical properties of Gabriel’s horn are the key inspiration for the legendary painter’s paradox. I should caution readers that the following content consists of mathematics that is typically suitable for Year 12 and Year 13 students with a suitable degree of mathematical dexterity. Volume of Gabriel’s Horn The volume of Gabriel’s horn can be calculated using the volumes of revolution formula: Therefore by substituting y=1/x and the limits x=1 and x=∞, we get: Thus, we have concluded mathematically that the volume of Gabriel’s horn is exactly ∩ cubic units. Gabriel’s Horn - S nkarshan Bhattacharyya Gabriel’s horn (also known as Torricelli’s trumpet ) is a hypothetical solid of revolution that o has infinite surface area but finite volume. It is modelled by the function = # $ ( 1 ≤ < ∞ ), rotated 360° or 2 radians around the -axis. The special mathematical properties of Gabri are the key inspiration for the legendary painter’s paradox . Figure I: Gabriel’s Horn I should caution readers that the following content consists of mathematics that is typically sui Year 12 and Year 13 students with a suitable degree of mathematical dexterity. Volume of Gabriel’s Horn The volume of Gabriel’s horn can be calculated using the volumes of revolution formula: = / % & ' d Therefore by substituting = # $ an the li its = 1 and = ∞ , we get: = / 0 1 1 % ( # d = / 1 % ( # d = / )% ( # d = 2 )# −1 4 # ( = 5− 1 6 # ( = − ∞ As the denominator of * ( gets increasingly larger, the value of * ( becomes increasingly smalle tends to a limiting value of zero (i.e., lim '→( : * ' ; = 0 ). This means that * ( = 0 . ∴ = − = RGSHW 2022/23 Academi Gabriel’s Horn - Sankarshan Bhattacharyya Gabriel’s horn (also known as Torricelli’s trumpet ) is a hypothetical solid of revolution that o has infinite surface area but finite volume. It is modelled by the function = # $ ( 1 ≤ < ∞ ), rotated 360° or 2 radians round the -axis. The special mathematical pr perties of Gabri are the key inspiration for the legendary painter’s paradox . Figure I: Gabriel’s Horn I should caution read rs that the f llowing content consists of mathematics that is typically sui Year 12 and Year 13 students with a suitable degree of mathematical dexterity. Volume of Gabriel’s Horn The volume of Gabriel’s horn can be calculated using the volumes of revolution formula: = / % & ' d Therefore by substit ting = # $ and the limits = 1 and = ∞ , we get: = / 0 1 1 % ( # d = / 1 % ( # d = / )% ( # d = 2 )# −1 4 # ( = 5− 1 6 # ( = − ∞ As the denominator of * ( gets increasingly larger, the value of * ( becomes increasingly smalle lim '→( : * ' ; = 0 * ( = 0 Figure 1: Gabriel’s Horn

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