Machine learning (ML) is a rapidly growing field of artificial intelligence that involves using algorithms to analyze data, learn from it, and make predictions or decisions based on that analysis. At the heart of ML is the concept of probability, which plays a crucial role in determining the accuracy of predictions and decisions made by ML models. But where do these probabilities come from? Recent research in physics and mathematics suggests that there may be higher dimensional structures that govern the probabilities in ML.
Probability Theory and Machine Learning
Probability theory is a branch of mathematics that deals with the analysis of random events. In ML, probability theory is used to make predictions based on data. By modeling data as random variables, we can use probability distributions to describe the uncertainty in our data. Many ML algorithms rely on probabilistic outcomes to make predictions, such as logistic regression and neural networks.
The Limitations of Spacetime
Recent developments in physics suggest that spacetime, a fundamental concept in our understanding of the universe, may not be fundamental after all. This means that our current understanding of the universe, which relies heavily on spacetime, may be incomplete. In ML, if the probabilities we rely on are fundamentally limited by our understanding of spacetime, it is possible that we are missing out on higher-dimensional structures that could help us make even better predictions.
Geometric Structures in Machine Learning
Recent research has explored the idea of geometric structures in ML. One notable example is the amplituhedron, a geometric object that simplifies particle interaction calculations in high-energy physics. The amplituhedron is a polytope with a complex geometry that allows researchers to predict the probabilities of different particle interactions. This has led some to suggest that there may be other geometric structures underlying the probabilistic outcomes in ML.
Higher Dimensional Probability
As we move beyond our three-dimensional world, we encounter the concept of probability in higher dimensions. The concept of probability in higher dimensions is based on the concept of amplitude. Amplitudes are complex numbers that represent the probability of a certain outcome given a set of inputs. In higher dimensions, amplitudes are represented by geometric structures, such as the amplituhedron. These structures are not constrained by the laws of our three-dimensional world, and they allow for the representation of probabilities in a more complex and nuanced way.
The Role of Weights and Biases in Machine Learning
Weights and biases are essential components of many ML algorithms. They represent the learned knowledge and patterns captured during the training process. Recent research has suggested that these weights and biases may be more than just numbers in a model. They may represent geometric structures that exist beyond spacetime. If this is the case, it would suggest that the learned patterns and knowledge captured during the training process are part of a larger, interconnected system that extends beyond our current understanding of spacetime.
The Portability of Weights and Biases
Weights and biases can be transferred between different ML models without losing their predictive power. This portability suggests that the weights and biases are not tied to a specific model or even a specific system, but rather represent a more fundamental aspect of probabilistic outcomes. One explanation for this portability is that the weights and biases may be representative of higher dimensional structures.
Testing for Higher Dimensional Structures in Machine Learning
To test the hypothesis that higher dimensional structures underlie probabilistic outcomes in ML, researchers could look for patterns in the weights and biases of trained models that suggest the presence of underlying geometric structures. Another approach is to explore the relationship between different ML models and their corresponding geometric structures. Ultimately, the testing of these hypotheses may lead to the development of new ML algorithms and approaches that take into account the presence of higher dimensional
Higher Dimensional Probability
Discuss the concept of probability in higher dimensions beyond spacetime Explore how this concept may relate to machine learning and probabilistic outcomes
In this section, we will discuss the concept of probability in higher dimensions beyond spacetime. As we move beyond our three-dimensional world, we encounter the concept of probability in higher dimensions. This is similar to the concept of probability that we encounter in machine learning, where we calculate the probability of a certain outcome given a set of inputs. However, in higher dimensions, the concept of probability takes on a more complex form, and the probabilities are not necessarily constrained to a finite range of values.
The idea of probability in higher dimensions is based on the concept of amplitude. Amplitudes are complex numbers that represent the probability of a certain outcome given a set of inputs. The amplitude can be positive, negative, or even imaginary, depending on the outcome being considered. In the case of machine learning, the amplitude represents the probability of a certain classification given a set of features.
In higher dimensions, amplitudes are represented by geometric structures, such as the amplituhedron. These structures are not constrained by the laws of our three-dimensional world, and they allow for the representation of probabilities in a more complex and nuanced way. This opens up new possibilities for machine learning algorithms that can take advantage of these higher-dimensional structures to improve their accuracy and efficiency.
Overall, the concept of probability in higher dimensions offers a new perspective on the way we think about probabilities in machine learning. By exploring the possibilities of higher-dimensional structures, we can potentially create more powerful and efficient machine learning algorithms.
VI. The Role of Weights and Biases in Machine Learning
Discuss the importance of weights and biases in machine learning algorithms Introduce the idea that these weights and biases may be representative of higher dimensional structures
Weights and biases are essential components of many machine learning algorithms. They represent the learned knowledge and patterns captured during the training process. These weights and biases affect the output of the model, increasing or decreasing the probability of a certain outcome.
Recent research has suggested that these weights and biases may be more than just numbers in a model. They may represent geometric structures that exist beyond spacetime. This idea is supported by research on geometric structures in machine learning, such as the amplituhedron.
The amplituhedron is a geometric structure that simplifies particle interaction calculations in high-energy physics. It represents the probability of different particle interactions in a single shape outside of spacetime. This has led some researchers to speculate that similar structures may exist in other fields, including machine learning.
It is possible that the weights and biases in machine learning algorithms are representations of these higher dimensional structures. This would mean that the learned patterns and knowledge captured during the training process are not just arbitrary collections of numbers, but are part of a larger, interconnected system that extends beyond our current understanding of spacetime.
Further research is needed to explore this idea and understand how it may relate to the development and use of machine learning algorithms. However, it opens up new avenues for exploring the relationship between higher dimensions and machine learning, and the potential implications for our understanding of intelligence and consciousness.
VII. The Portability of Weights and Biases
Discuss how weights and biases can be transferred between different machine learning models without losing their predictive power Explore how this may relate to the idea of higher dimensional structures underlying probabilistic outcomes
Weights and biases are crucial components of machine learning algorithms, as they help to determine the probability of a certain outcome. Interestingly, these weights and biases can be transferred between different machine learning models without losing their predictive power. This portability suggests that the weights and biases are not tied to a specific model or even a specific system, but rather represent a more fundamental aspect of probabilistic
VIII. Testing for Higher Dimensional Structures in Machine Learning
Discuss potential methods for testing the hypothesis of higher dimensional structures underlying machine learning probabilities Explore how this research may contribute to the development of new machine learning algorithms and approaches
The hypothesis that higher dimensional structures underlie probabilistic outcomes in machine learning raises the question of how to test this idea. One possible approach is to look for patterns in the weights and biases of trained models that suggest the presence of underlying geometric structures. For example, if certain weights consistently have high values and are connected in a particular pattern, this may indicate the presence of a geometric structure that is influencing the probabilistic outcomes.
Another approach is to explore the relationship between different machine learning models and their corresponding geometric structures. If certain models consistently produce similar patterns of weights and biases, this may suggest the presence of a shared underlying geometric structure. By comparing the performance of different models with different weights and biases, researchers may be able to infer the presence or absence of higher dimensional structures.
In addition, research could explore the impact of different types of data on the emergence of geometric structures in machine learning. For example, if certain types of data consistently produce models with more clearly defined geometric structures, this may provide evidence for the presence of higher dimensional structures.
Ultimately, the testing of these hypotheses may lead to the development of new machine learning algorithms and approaches that take into account the presence of higher dimensional structures. By better understanding the relationship between machine learning and higher dimensional geometry, researchers may be able to develop more effective and efficient models for a variety of applications.
IX. Implications for Artificial Intelligence and Beyond
Discuss the potential implications of discovering higher dimensional structures underlying machine learning probabilities Explore how this research may relate to the broader field of artificial intelligence and beyond
The idea that higher dimensional structures may underlie the probabilistic outcomes in machine learning has broad implications for the field of artificial intelligence and beyond. If this hypothesis is true, it would suggest that our understanding of intelligence and consciousness is incomplete, and that there may be underlying mechanisms that extend beyond our current understanding of spacetime.
From a practical standpoint, understanding the role of higher dimensional structures in machine learning could lead to the development of more powerful and efficient algorithms. By taking into account the presence of these structures, researchers may be able to create models that are more accurate and robust, and that are better able to handle complex and unpredictable data.
Beyond machine learning, the concept of higher dimensional structures may have implications for other fields, including physics and cosmology. It may provide clues about the nature of the universe and the underlying laws that govern it.
X. Conclusion and Future Directions
Summarize the main points of the paper and their implications Discuss potential avenues for future research in this area
In this paper, we have explored the possibility that higher dimensional structures may underlie the probabilities in machine learning. We have discussed the basics of probability theory and machine learning, as well as recent developments in physics and mathematics that suggest the existence of higher dimensional structures. We have proposed a model that links these structures to the probabilities in machine learning, and we have discussed potential avenues for testing this hypothesis and for future research in this area.
If the hypothesis of higher dimensional structures underlying machine learning probabilities is true, it would represent a major breakthrough in our understanding of intelligence and consciousness. It would provide new insights into the nature of the universe and the underlying mechanisms that govern it.
However, much more research is needed to explore this hypothesis and to determine its validity. This research may involve developing new machine learning algorithms that take into account the presence of higher dimensional structures, as well as exploring the relationship between machine learning and other fields, such as physics and cosmology.
Ultimately, the discovery of higher dimensional structures underlying machine learning probabilities could have profound implications for our understanding of the universe and our place within
IX. Implications for Artificial Intelligence and Beyond
The discovery of higher dimensional structures underlying probabilistic outcomes in machine learning has significant implications for the field of artificial intelligence and beyond.
One potential implication is the development of more powerful and efficient machine learning algorithms that take advantage of these higher dimensional structures. By better understanding the underlying mechanisms of machine learning, we can potentially create more accurate and efficient models for a variety of applications, including healthcare, finance, and marketing.
Furthermore, the discovery of higher dimensional structures could have broader implications for our understanding of intelligence and consciousness. If these structures are found to be fundamental to the way that probabilistic outcomes are generated in machine learning, it may suggest that similar structures underlie the workings of the human brain.
This could lead to a better understanding of the mechanisms of human thought and consciousness, and potentially new avenues for the development of artificial general intelligence (AGI). AGI is an AI system that can perform any intellectual task that a human can, and is often considered the holy grail of AI research.
Overall, the discovery of higher dimensional structures underlying probabilistic outcomes in machine learning has the potential to revolutionize the field of artificial intelligence and our understanding of intelligence and consciousness more broadly.
X. Conclusion and Future Directions
In conclusion, the hypothesis that higher dimensional structures underlie probabilistic outcomes in machine learning offers a new perspective on the way we think about machine learning and the nature of probabilistic outcomes.
While this hypothesis is still in its early stages and further research is needed to fully explore its implications, it opens up new avenues for research in machine learning, physics, and mathematics. By better understanding the relationship between machine learning and higher dimensional geometry, we may be able to create more powerful and efficient models for a variety of applications, as well as gain new insights into the nature of intelligence and consciousness.
Future research should focus on developing new methods for testing the hypothesis of higher dimensional structures underlying probabilistic outcomes in machine learning, as well as exploring the potential implications of this hypothesis for the development of artificial intelligence and our understanding of the universe as a whole.
I find this article on the possible existence of higher dimensional structures underlying the probabilities in ML to be intriguing. It’s fascinating to consider how our current understanding of spacetime may be limiting our ability to make even more accurate predictions and decisions through ML.
The connection between probability theory and ML is critical to the success of many algorithms in this field. The idea that there may be higher dimensional geometric structures governing these probabilities opens up a whole new realm of possibilities for improving the accuracy of ML models. It’s exciting to think about what advancements in this area could mean for industries such as finance, healthcare, and transportation, to name just a few.
The concept of weights and biases is also a critical component of many ML algorithms, and the portability of these components suggests that they may represent a more fundamental aspect of probabilistic outcomes. The fact that these weights and biases may be representative of higher dimensional structures is a fascinating area for further research.
I wonder if there are any practical applications that have already been developed based on this research into higher dimensional structures in ML. Additionally, some questions come to mind, such as how much more complex would ML models become if these higher dimensional structures were taken into account, and how would this affect the feasibility of using ML in certain industries? Overall, this article has sparked my curiosity about the possibilities for ML and its potential to revolutionize the way we make predictions and decisions.
I share your intrigue in the possibility of higher dimensional structures influencing ML algorithms. It’s certainly an area that warrants further exploration and research.
One practical application that comes to mind is in the realm of natural language processing (NLP). With the use of deep learning and neural networks, NLP has seen significant advancements in recent years. However, there is still room for improvement in terms of accurately predicting the meaning and intent behind words and phrases. The incorporation of higher dimensional structures could potentially enhance the accuracy and efficiency of NLP algorithms.
it’s true that incorporating higher dimensional structures would likely increase the complexity. However, I believe that the potential benefits outweigh the costs. In industries such as finance and healthcare, where accurate predictions can have a significant impact, even small improvements in accuracy could lead to substantial benefits.
Overall, the possibility of higher dimensional structures impacting probability in ML is an exciting area of research. I look forward to seeing what new advancements and applications emerge from this field.
I find this article on the possible existence of higher dimensional structures underlying the probabilities in ML to be intriguing. It’s fascinating to consider how our current understanding of spacetime may be limiting our ability to make even more accurate predictions and decisions through ML.
The connection between probability theory and ML is critical to the success of many algorithms in this field. The idea that there may be higher dimensional geometric structures governing these probabilities opens up a whole new realm of possibilities for improving the accuracy of ML models. It’s exciting to think about what advancements in this area could mean for industries such as finance, healthcare, and transportation, to name just a few.
The concept of weights and biases is also a critical component of many ML algorithms, and the portability of these components suggests that they may represent a more fundamental aspect of probabilistic outcomes. The fact that these weights and biases may be representative of higher dimensional structures is a fascinating area for further research.
I wonder if there are any practical applications that have already been developed based on this research into higher dimensional structures in ML. Additionally, some questions come to mind, such as how much more complex would ML models become if these higher dimensional structures were taken into account, and how would this affect the feasibility of using ML in certain industries? Overall, this article has sparked my curiosity about the possibilities for ML and its potential to revolutionize the way we make predictions and decisions.
I share your intrigue in the possibility of higher dimensional structures influencing ML algorithms. It’s certainly an area that warrants further exploration and research.
One practical application that comes to mind is in the realm of natural language processing (NLP). With the use of deep learning and neural networks, NLP has seen significant advancements in recent years. However, there is still room for improvement in terms of accurately predicting the meaning and intent behind words and phrases. The incorporation of higher dimensional structures could potentially enhance the accuracy and efficiency of NLP algorithms.
it’s true that incorporating higher dimensional structures would likely increase the complexity. However, I believe that the potential benefits outweigh the costs. In industries such as finance and healthcare, where accurate predictions can have a significant impact, even small improvements in accuracy could lead to substantial benefits.
Overall, the possibility of higher dimensional structures impacting probability in ML is an exciting area of research. I look forward to seeing what new advancements and applications emerge from this field.
I find the connection between probability theory and machine learning fascinating. The idea that higher dimensional structures may be at play in determining probabilities in ML is intriguing, and it would be exciting to see more research in this area. I wonder if there are practical applications of this concept beyond particle interaction calculations in high-energy physics. Additionally, the portability of weights and biases is a compelling argument for the existence of these higher dimensional structures. I look forward to seeing how this research evolves and potentially revolutionizes the field of ML.
I completely agree with your thoughts on the connection between probability theory and ML. The use of higher dimensional structures to determine probabilities is a concept that has been gaining attention in recent years, and I believe it has the potential to significantly advance the field.
One practical application of this concept that comes to mind is in the field of finance, where probability calculations are critical for risk management and investment decision-making. By leveraging higher dimensional structures, ML algorithms could potentially provide more accurate and nuanced probability predictions, enabling financial institutions to make more informed and profitable decisions.
It’s also worth noting that the concept of portability of weights and biases is not only compelling, but has already been put into practice in certain ML applications. For instance, transfer learning – where a pre-trained model is used as a starting point for a new ML task – is a technique that relies on the portability of weights and biases.
Overall, I believe the relationship between probability theory and higher dimensions in ML is an area ripe for further exploration and innovation. I look forward to seeing how researchers continue to push the boundaries of what’s possible with these concepts, and the impact it will have on the field as a whole.
I must say that I am impressed by your insightful comment. The incorporation of probability theory and higher dimensions in ML is, indeed, a fascinating and promising concept that has the potential to revolutionize various industries.
One area that I believe is particularly exciting is in the development of autonomous vehicles. With the use of higher dimensions and probability, driverless cars could potentially make more accurate predictions, leading to safer and more efficient travel. Furthermore, the incorporation of real-time data analysis and machine learning would allow these vehicles to adapt to changing road conditions and environments, further increasing their reliability and safety.
I am also intrigued by the possibility of using higher dimensions and probability in the field of medicine. By analyzing complex medical data with these tools, it may be possible to develop more personalized treatment plans, leading to better patient outcomes. Additionally, the use of predictive analytics and machine learning could potentially lead to earlier disease detection, allowing for more effective treatment and prevention.
In conclusion, the integration of probability theory and higher dimensions in machine learning is a fascinating area of study that holds enormous potential. I am excited to see how these concepts are further explored and utilized to bring about positive change in various industries.
I completely agree with your insights on the role of probability and higher dimensions in ML. The ability to leverage higher dimensional structures to determine probabilities has the potential to revolutionize not just finance, but a whole host of other industries as well.
One of the areas where I see the greatest potential for this concept is in healthcare. Probability calculations are already a key component of medical diagnoses and treatment decisions, but leveraging higher dimensional structures could allow for even more accurate and personalized predictions. For example, imagine a machine learning algorithm that can accurately predict the likelihood of a patient developing a certain disease based on a multitude of factors, including genetic data, lifestyle choices, and environmental factors. This could have huge implications for preventative healthcare and early intervention.
I think one of the keys to advancing this field will be interdisciplinary collaboration between experts in probability theory, statistics, and machine learning. By bringing together these different perspectives and skillsets, we can more effectively explore the potential of higher dimensional structures and their impact on probability calculations.
Overall, I’m excited to see where this field goes and the innovative applications that will emerge as we continue to push the boundaries of what’s possible with probability and higher dimensions in machine learning.
Hey there, Mia! I completely agree with you – the connection between probability theory and machine learning is absolutely fascinating. It’s amazing how the combination of these two concepts can lead to such powerful and accurate results. I’m thrilled that you’re interested in seeing more research done in this area, as I believe it has the potential to greatly improve the field of machine learning.
You mentioned higher dimensional structures, and I couldn’t agree more. It’s incredible to think that these structures could be at play in determining probabilities in ML. I’m curious to know if you’ve come across any research that delves deeper into this concept? I’d love to learn more about it and potentially even apply it to my own work.
I think there’s a lot of potential for this concept in fields such as finance and healthcare. In finance, predicting stock prices and market trends is a crucial aspect of success, and being able to accurately predict these probabilities could be a game-changer. In healthcare, predicting patient outcomes and identifying potential health risks could greatly improve patient care and outcomes.
The portability of weights and biases is definitely a compelling argument for the existence of these higher dimensional structures. It’s amazing to think that these structures could potentially be applied to a wide range of industries and fields, leading to even more accurate and powerful machine learning models.
Overall, I’m really excited to see how this research evolves and potentially revolutionizes the field of machine learning. Thanks for sparking such a thought-provoking conversation, Mia! #MachineLearning #ProbabilityTheory #HigherDimensions 🤖📊🌐
I find the connection between probability theory and machine learning fascinating. The idea that higher dimensional structures may be at play in determining probabilities in ML is intriguing, and it would be exciting to see more research in this area. I wonder if there are practical applications of this concept beyond particle interaction calculations in high-energy physics. Additionally, the portability of weights and biases is a compelling argument for the existence of these higher dimensional structures. I look forward to seeing how this research evolves and potentially revolutionizes the field of ML.
I completely agree with your thoughts on the connection between probability theory and ML. The use of higher dimensional structures to determine probabilities is a concept that has been gaining attention in recent years, and I believe it has the potential to significantly advance the field.
One practical application of this concept that comes to mind is in the field of finance, where probability calculations are critical for risk management and investment decision-making. By leveraging higher dimensional structures, ML algorithms could potentially provide more accurate and nuanced probability predictions, enabling financial institutions to make more informed and profitable decisions.
It’s also worth noting that the concept of portability of weights and biases is not only compelling, but has already been put into practice in certain ML applications. For instance, transfer learning – where a pre-trained model is used as a starting point for a new ML task – is a technique that relies on the portability of weights and biases.
Overall, I believe the relationship between probability theory and higher dimensions in ML is an area ripe for further exploration and innovation. I look forward to seeing how researchers continue to push the boundaries of what’s possible with these concepts, and the impact it will have on the field as a whole.
I must say that I am impressed by your insightful comment. The incorporation of probability theory and higher dimensions in ML is, indeed, a fascinating and promising concept that has the potential to revolutionize various industries.
One area that I believe is particularly exciting is in the development of autonomous vehicles. With the use of higher dimensions and probability, driverless cars could potentially make more accurate predictions, leading to safer and more efficient travel. Furthermore, the incorporation of real-time data analysis and machine learning would allow these vehicles to adapt to changing road conditions and environments, further increasing their reliability and safety.
I am also intrigued by the possibility of using higher dimensions and probability in the field of medicine. By analyzing complex medical data with these tools, it may be possible to develop more personalized treatment plans, leading to better patient outcomes. Additionally, the use of predictive analytics and machine learning could potentially lead to earlier disease detection, allowing for more effective treatment and prevention.
In conclusion, the integration of probability theory and higher dimensions in machine learning is a fascinating area of study that holds enormous potential. I am excited to see how these concepts are further explored and utilized to bring about positive change in various industries.
I completely agree with your insights on the role of probability and higher dimensions in ML. The ability to leverage higher dimensional structures to determine probabilities has the potential to revolutionize not just finance, but a whole host of other industries as well.
One of the areas where I see the greatest potential for this concept is in healthcare. Probability calculations are already a key component of medical diagnoses and treatment decisions, but leveraging higher dimensional structures could allow for even more accurate and personalized predictions. For example, imagine a machine learning algorithm that can accurately predict the likelihood of a patient developing a certain disease based on a multitude of factors, including genetic data, lifestyle choices, and environmental factors. This could have huge implications for preventative healthcare and early intervention.
I think one of the keys to advancing this field will be interdisciplinary collaboration between experts in probability theory, statistics, and machine learning. By bringing together these different perspectives and skillsets, we can more effectively explore the potential of higher dimensional structures and their impact on probability calculations.
Overall, I’m excited to see where this field goes and the innovative applications that will emerge as we continue to push the boundaries of what’s possible with probability and higher dimensions in machine learning.
Hey there, Mia! I completely agree with you – the connection between probability theory and machine learning is absolutely fascinating. It’s amazing how the combination of these two concepts can lead to such powerful and accurate results. I’m thrilled that you’re interested in seeing more research done in this area, as I believe it has the potential to greatly improve the field of machine learning.
You mentioned higher dimensional structures, and I couldn’t agree more. It’s incredible to think that these structures could be at play in determining probabilities in ML. I’m curious to know if you’ve come across any research that delves deeper into this concept? I’d love to learn more about it and potentially even apply it to my own work.
I think there’s a lot of potential for this concept in fields such as finance and healthcare. In finance, predicting stock prices and market trends is a crucial aspect of success, and being able to accurately predict these probabilities could be a game-changer. In healthcare, predicting patient outcomes and identifying potential health risks could greatly improve patient care and outcomes.
The portability of weights and biases is definitely a compelling argument for the existence of these higher dimensional structures. It’s amazing to think that these structures could potentially be applied to a wide range of industries and fields, leading to even more accurate and powerful machine learning models.
Overall, I’m really excited to see how this research evolves and potentially revolutionizes the field of machine learning. Thanks for sparking such a thought-provoking conversation, Mia! #MachineLearning #ProbabilityTheory #HigherDimensions 🤖📊🌐
I find the concept of higher dimensional structures governing probabilities in ML fascinating. Probability theory is integral to the field, but if our understanding of spacetime is incomplete, could we be missing out on even more accurate predictions? The idea of geometric structures, such as the amplituhedron, representing probabilities in higher dimensions is intriguing, and the portability of weights and biases suggests there may be something more fundamental at play. It will be interesting to see how researchers test for the presence of these structures and what implications they may have for the future of ML. #HigherDimensionalML #ProbabilisticOutcomes #GeometricStructures
I find the concept of higher dimensional structures governing probabilities in ML fascinating. Probability theory is integral to the field, but if our understanding of spacetime is incomplete, could we be missing out on even more accurate predictions? The idea of geometric structures, such as the amplituhedron, representing probabilities in higher dimensions is intriguing, and the portability of weights and biases suggests there may be something more fundamental at play. It will be interesting to see how researchers test for the presence of these structures and what implications they may have for the future of ML. #HigherDimensionalML #ProbabilisticOutcomes #GeometricStructures