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2021 | OriginalPaper | Chapter

Visualizing Modular Forms

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Abstract

We examine several currently used techniques for visualizing complex-valued functions applied to modular forms. We plot several examples and study the benefits and limitations of each technique. We then introduce a method of visualization that can take advantage of colormaps in Python’s matplotlib library, describe an implementation, and give more examples. Much of this discussion applies to general visualizations of complex-valued functions in the plane.
Footnotes
1
Adjusting the map from magnitude to brightness for SageMath’s complex_plot is possible, but nontrivial; the map is defined internally and the current plotting interface doesn’t give any option to choose or alter this map. In order to adjust this map, it is necessary to modify the source for SageMath’s plotting routines directly.
 
2
In the period between writing and publishing this paper, the LMFDB changed its method for plotting modular forms to incorporate methods described in this paper. Thus in the rest of this paper, LMFDB-style plots refer to the style of plots in the LMFDB on October 1, 2020.
 
3
The behavior in Matlab and Maple appears similar, though it is possible to define a color scheme. On the other hand Mathematica has an extensive library of complex plotting color schemes.
 
Literature
[DS05]
go back to reference Fred Diamond and Jerry Shurman. A first course in modular forms, volume 228. Springer Verlag, 2005. MATH Fred Diamond and Jerry Shurman. A first course in modular forms, volume 228. Springer Verlag, 2005. MATH
[Far98]
go back to reference Frank A Farris. Visual complex analysis by Tristan Needham. The American Mathematical Monthly, 105(6):570–576, 1998. Frank A Farris. Visual complex analysis by Tristan Needham. The American Mathematical Monthly, 105(6):570–576, 1998.
[Far15]
go back to reference Frank A Farris. Creating Symmetry: The artful mathematics of wallpaper patterns. Princeton University Press, 2015. Frank A Farris. Creating Symmetry: The artful mathematics of wallpaper patterns. Princeton University Press, 2015.
[Hun07]
go back to reference J. D. Hunter. Matplotlib: A 2d graphics environment. Computing in Science & Engineering, 9(3):90–95, 2007. CrossRef J. D. Hunter. Matplotlib: A 2d graphics environment. Computing in Science & Engineering, 9(3):90–95, 2007. CrossRef
[Kov15]
go back to reference Peter Kovesi. Good colour maps: How to design them. arXiv preprint arXiv:1509.03700, 2015. Peter Kovesi. Good colour maps: How to design them. arXiv preprint arXiv:1509.03700, 2015.
[NAR18]
go back to reference Jamie R Nuñez, Christopher R Anderton, and Ryan S Renslow. Optimizing colormaps with consideration for color vision deficiency to enable accurate interpretation of scientific data. PloS one, 13(7):e0199239, 2018. Jamie R Nuñez, Christopher R Anderton, and Ryan S Renslow. Optimizing colormaps with consideration for color vision deficiency to enable accurate interpretation of scientific data. PloS one, 13(7):e0199239, 2018.
[Oli06]
go back to reference Travis E Oliphant. A guide to NumPy, volume 1. Trelgol Publishing USA, 2006. Travis E Oliphant. A guide to NumPy, volume 1. Trelgol Publishing USA, 2006.
[WS10]
go back to reference Elias Wegert and Gunter Semmler. Phase plots of complex functions: a journey in illustration. Notices AMS, 58:768–780, 2010. MathSciNetMATH Elias Wegert and Gunter Semmler. Phase plots of complex functions: a journey in illustration. Notices AMS, 58:768–780, 2010. MathSciNetMATH
Metadata
Title
Visualizing Modular Forms
Author
David Lowry-Duda
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-80914-0_19

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