class: center, middle, inverse, title-slide .title[ #
Advancing Statistical Quality Control with AI: Insights from ChatGPT and ChatSQC
] .author[ ### Ying-Ju Tessa Chen, PhD
Scholar
|
@ying-ju
|
ychen4@udayton.edu
Joint work with:
Fadel Megahed, PhD
Miami University
Inez Zwetsloot, PhD
University of Amsterdam
Sven Knoth, PhD
Helmut-Schmidt-Universität
Douglas C. Montgomery, PhD
Arizona State University
Allison Jones-Farmer, PhD
Miami University
] .date[ ### February 11, 2024 | Math Club | Dayton OH ] --- # Background: Artificial Intelligence .left-code[ .center[.bold[A [working definition](https://www.brookings.edu/articles/what-is-artificial-intelligence/) for AI]] .content-box-gray[ .bold[.red[Artificial Intelligence (AI):]] .bold[A system that acts in a way, where people might denote as "intelligent" if another human were to do something similar.] ] .center[.bold[Reported applications of AI span numerous fields]] e.g., see a sample applications as generated by ChatGPT in the flowchart to the right. ] .right-plot[ <img src="figs/ai_applications.png" alt="A flowchart highlighting the applications of AI, with highlight in red for fraud detection (in e-commerce), grading and assessment (in education), quality control and predictive maintenance (in manufacturing and production) as they relate the most to our SPC audience" width="100%" style="display: block; margin: auto;" /> ] .footnote[ <html> <hr> </html> **Image Source:** The flowchart's content and its LaTex code were generated using ChatGPT (May 24 Version). ] --- # Background: Generative AI .content-box-gray[ .bold[.red[Generative AI:]] .bold[The objective is to generate new content rather than analyze existing data.] ] .font90[ - The generated content is based on a .bold[.red[stochastic behavior embedded in generative AI models such that the same input prompts results in different content]]. - State-of-the-art generative AI models can have up to **540 billion parameters** ([PaLM](https://arxiv.org/abs/2204.02311)). - With the increase in model size, researchers have observed the **“emergent abilities”** of LLMs, which were **not explicitly encoded in the training**. [Examples include](https://ai.googleblog.com/2022/11/characterizing-emergent-phenomena-in.html): + Multi-step arithmetic, + taking college-level exams, and + identifying the intended meaning of a word. - LLMs are **foundation models** (see [Bommasani et al. 2021](https://arxiv.org/abs/2108.07258)), large pre-trained AI systems that can be **repurposed with minimal effort across numerous domains and diverse tasks.** ] --- # Background: Generative AI Developments <img src="figs/ai_dev.png" width="100%" style="display: block; margin: auto;" /> --- # Background: Generative AI Hype <img src="figs/chatgpt_hype.png" alt="It took ChatGPT 5 days to reach 1 million users, while Instagram took 2.5 months and twitter took 24 months. We have not seen such a huge adoption that quick in tech before." width="100%" style="display: block; margin: auto;" /> --- # Background: Generative AI Hype <img src="figs/mckinsey_ai.png" width="60%" style="display: block; margin: auto;" /> .footnote[ <html> <hr> </html> **Image Source:** [McKinsey & Company (July 2023). The economic potential of generative AI: The next productivity frontier [P. 10]](https://www.mckinsey.com/~/media/mckinsey/business%20functions/mckinsey%20digital/our%20insights/the%20economic%20potential%20of%20generative%20ai%20the%20next%20productivity%20frontier/the-economic-potential-of-generative-ai-the-next-productivity-frontier-vf.pdf) ] --- class: inverse, center, middle # On the Use of LLMs, such as ChatGPT, in SQC <br> .pull-left-2[<br> Megahed, F. M., Chen, Y. J., Ferris, J. A., Knoth, S., & Jones-Farmer, L. A. (2023). How generative AI models such as ChatGPT can be (mis)used in SPC practice, education, and research? An exploratory study. *Quality Engineering*, 1-29. [Available @ [QE](https://www.tandfonline.com/doi/full/10.1080/08982112.2023.2206479) & [arXiv](https://arxiv.org/pdf/2302.10916.pdf)].] .pull-right-2[<div><img src="figs/paper_qr_code.png" class="paper-img" width="300px" align="right"></div>] --- # Our Overarching Research Question .content-box-red[ .bold[What can generative LLM-based AI tools do now to augment the roles of SPC practitioners, educators, and researchers?] ] - **Secondary goal:** To motivate the SPC community to be receptive to exploring whether new AI tools can help them be more **efficient**, **productive**, and **innovative**. This is consistent with: + Box and Woodall ([2012](https://www.tandfonline.com/doi/10.1080/08982112.2012.627003)): “we stress the necessity for the quality engineering community to strengthen and promote its role in **innovation**”, and + Hockman and Jensen ([2016](https://www.tandfonline.com/doi/10.1080/08982112.2015.1083107)): “for statisticians to be successful in leading innovation, they will need to strengthen their **skills beyond what they have traditionally needed in the past**, but we believe this will be worth the effort”. - **Scope:** We evaluated the utility of ChatGPT (GPT-3.5 engine) as of its *Jan 30 Version*. --- # Our Study Design <img src="./figs/methods.png" width="68%" style="display: block; margin: auto;" /> --- # The Good: Knowledge Generation .bold[Inspired by the TEDxBoston talk titled [what we learned from 5 million books](https://www.ted.com/talks/jean_baptiste_michel_erez_lieberman_aiden_what_we_learned_from_5_million_books?language=en), we asked ChatGPT the following question:] <br> > .bold[.large["What are open issues in statistical process control research?'']] <br> ### Why this question seemed like a reasonable prompt? .bold[ChatGPT likely “read” and “can recall” more SPC research papers than most of us] --- # The Good: Knowledge Generation <img src="figs/research_prompt_08_fig_01.png" alt="Chat GPT highlighted six areas where there are open issues in statistical process control. We will highlight the main themes in the next slide" width="60%" style="display: block; margin: auto;" /> --- # The Good: Knowledge Generation .content-box-red[ .center[.bold[.large[Some Thoughts on the ChatGPT Answer]]] - It captured .bold[reasonable themes, e.g., ] + incorporating .bold[big data and machine learning] techniques, + .bold[online/real-time monitoring] solutions where 100% sampling is employed, + the need for .bold[non-normality], and + .bold[applications to new domains]. - In our opinion, .bold[value is in using it as a high-level tool for idea generation/validation]. - Potentially .bold[“stale”] as [Chat(GPT)-3.5 “finished training in early 2022”](https://openai.com/blog/chatgpt/) and is limited to [data up to Sept 2021](https://community.openai.com/t/knowledge-cutoff-date-of-september-2021/66215). + Probably not an issue for future LLM generations (.bold[Why?]) ] --- # The Bad: Precise Definitions <img src="figs/research_prompt_05_fig_01.png" alt="ChatGPT's generated response for our prompt of explain the practitioner-to-practitioner variability. Its response is somewhat long and imprecise. Specifically, ChatGPT presented five factors, which share a common feature; all deal with differences on the method level, i.e., chart type, subgroup design, techniques to calculate the limits, dealing with outliers, and choice of software. While we agree that these factors are important and will drive different results, ChatGPT's answer ignores the context for which the practitioner-to-practitioner variability is used in the SPC literature. In fact, the practitioner-to-practitioner variability refers to the variation that occurs with a fixed configuration of the five aforementioned factors, i.e., the variation results from multiple implementations of the same procedure on the same data-generating process." width="43%" style="display: block; margin: auto;" /> --- # The Ugly: ChatGPT's Hallucination .bold[To detect whether ChatGPT can detect erroneous requests, we asked:] <br> > .bold[.large["Can you use the ‘bigfish' dataset from the qcc library in R to create a control chart?'']] <br> ### Why this question seemed like a reasonable prompt? .bold[In an earlier question (within the same thread), ChatGPT answered a question by using the `qcc` package, i.e., is .red[familiar with it], and .red[detecting unreasonable requests would be a strong feature for non-expert users].] --- # The Ugly: ChatGPT's Hallucination <img src="figs/practice_prompt_03_fig_01.png" alt="The ChatGPT hallucination, answering a question about a non-existent dataset in the qcc library" width="55%" style="display: block; margin: auto;" /> --- # The Ugly: ChatGPT's Hallucination <img src="figs/practice_prompt_03_fig_02.png" alt="ChatGPT making up details about the non-existent bigfish dataset and saying it is popular in the SPC community" width="80%" style="display: block; margin: auto;" /> --- class: inverse, center, middle # ChatSQC: Our Grounded App, to address Imprecise SQC Answers and Hallucinations <br> Megahed, F. M., Chen, Y. J., Zwetsloot, I., Knoth, S., Montgomery, D.C., & Jones-Farmer, L. A. (2023). AI and the Future of Work in Statistical Quality Control: Insights from a First Attempt to Augmenting ChatGPT with an SQC Knowledge Base (ChatSQC). Under Review. --- # The Construction of ChatSQC <img src="figs/version3.jpg" alt="The construction of ChatSQC involved four main phases: (a) a one-time extraction of the reference material, (b) a one-time preprocessing of the extracted material, (c) a continuous (online) chat inference, and (d) the hosting/deployment of the app on a web server." width="80%" style="display: block; margin: auto;" /> --- # A Live Demo of ChatSQC <center> <a href="https://chatsqc.osc.edu/"> <img alt="The interface to our ChatSQC app" src="figs/chatsqc_demo.png" style="width:100%; height:100%;"> </a> </center> --- # Concluding Remarks .font90[ - **Educational Grounding:** *ChatSQC* uniquely grounds its answers in the [NIST/SEMATECH e-handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/index.htm), ensuring users receive authoritative and educational responses. - **Transparency in Answers:** Instead of merely serving answers, we provide links, snapshots of the five most relevant text passages (along with their L2 distances to the prompt), and the costs incurred to generate these responses. - **Enhanced Privacy:** By utilizing the OpenAI API: [At OpenAI, protecting user data is fundamental to our mission. We do not train our models on inputs and outputs through our API](https://openai.com/api-data-privacy). - Our **Roadmap:** + Utilizing open-source LLMs. + Working with Wiley to get permission to include copyrighted materials such as [Introduction to Statistical Quality Control](https://www.wiley.com/en-us/Introduction+to+Statistical+Quality+Control%2C+8th+Edition-p-9781119399308). + Incorporating multiple sources/retrievers in *ChatSQC*. + Incorporating additional feedback from the community. ] --- ## Thank You! .pull-left[ - This presentation was created based on Dr. Fadel Megahed's presentation at [ISQC 2023](https://www.american.edu/cas/isqc/). Click [here](https://fmegahed.github.io/talks/isqc2023/chatsqc.html) to find the original presentation. - Please do not hesitate to contact me (Tessa Chen) at <a href="mailto:ychen@udayton.edu"><i class="fa fa-paper-plane fa-fw"></i> ychen4@udayton.edu</a> for questions or further discussions. ] .pull-right[ <img src="./figs/Tessa_grey_G.gif" width="60%" style="display: block; margin: auto;" /> ] --- class: center, middle, inverse, title-slide .title[ # <p>Advancing Statistical Quality Control with AI: Insights from ChatGPT and ChatSQC</p> ] .author[ ### Ying-Ju Tessa Chen, PhD <br>[<svg viewBox="0 0 488 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M488 261.8C488 403.3 391.1 504 248 504 110.8 504 0 393.2 0 256S110.8 8 248 8c66.8 0 123 24.5 166.3 64.9l-67.5 64.9C258.5 52.6 94.3 116.6 94.3 256c0 86.5 69.1 156.6 153.7 156.6 98.2 0 135-70.4 140.8-106.9H248v-85.3h236.1c2.3 12.7 3.9 24.9 3.9 41.4z"></path></svg> Scholar](https://scholar.google.com/citations?user=nfXnYKcAAAAJ&hl=en&oi=ao) | [<svg viewBox="0 0 496 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"></path></svg> @ying-ju](https://github.com/ying-ju) | [<svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"></path></svg>ychen4@udayton.edu](mailto:ychen4@udayton.edu)</br><br><u><b><font color="white">Joint work with:</b></u><br>Fadel Megahed, PhD [<svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"></path></svg> Miami University](https://miamioh.edu/fsb/directory/?up=/directory/megahefm)<br/>Inez Zwetsloot, PhD [<svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"></path></svg> University of Amsterdam](https://www.uva.nl/en/profile/z/w/i.m.zwetsloot/i.m.zwetsloot.html)<br/>Sven Knoth, PhD [<svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"></path></svg> Helmut-Schmidt-Universität](https://www.hsu-hh.de/compstat/en/sven-knoth-2)<br/>Douglas C. Montgomery, PhD [<svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"></path></svg> Arizona State University](https://search.asu.edu/profile/10123)<br>Allison Jones-Farmer, PhD [<svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"></path></svg> Miami University](https://miamioh.edu/fsb/directory/?up=/directory/farmerl2)<br><br/> ] .date[ ### February 12, 2024 | Math Club | Dayton OH ]