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Packable hyperbolic surfaces with symmetries
Part of
Maria Dostert, Alexander Kolpakov
Journal:

Canadian Mathematical Bulletin / Volume 66 / Issue 1 / March 2023

Published online by Cambridge University Press:

11 February 2022, pp. 103-113

Print publication:

March 2023
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We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof).

8 - Other Flat-Folding Problems
from Part II - The Combinatorial Geometry of Flat Origami
Thomas C. Hull
Book:

Origametry

Published online:

06 October 2020

Print publication:

08 October 2020, pp 159-178
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Summary

Chapter 8 covers a variety of other mathematical problems that arise from flat origami.For example, how many times can you fold a piece of paper in half?This old question has a more interesting answer, discovered by a high-school student, than is commonly known.Can any shape be folded?Origami artists want to claim that the answer to this question is, “Yes,” but how should the question be formalized mathematically?And if we really can fold anything, how do we do so?The process of origami design can be quantified, and the ways in which complex origami designers do so to, say, create an origami insect are summarized.Another origami puzzle with a surprising answer is:If a square piece of paper is folded into a flat object, what is the biggest perimeter that object can have?Finally, we summarize the theory behind, and proof of, the famous Fold-and-One-Cut Theorem.

13 - The Noise Case
from Part II - Geometry and Statistics
Patrick Flandrin
Book:

Explorations in Time-Frequency Analysis

Published online:

22 August 2018

Print publication:

06 September 2018, pp 116-123
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Summary

When applied to white Gaussian noise, Gaussian spectrograms are made of patches whose distribution is controlled by the correlation function of the STFT considered as a 2D homogeneous field, and in turn by the underlying reproducing kernel of the analysis. This can be given a simple model whose basic ingredients are a mean distribution of logons resulting from a circle packing argument due to uncertainty, and an adequate degree of randomness in fluctuations around the mean model and inter-logon phase relationships.

Asymptotic Behavior of Optimal Circle Packings in a Square
Kari J. Nurmela, Patric R. J. Östergård, Rainer aus dem Spring
Journal:

Canadian Mathematical Bulletin / Volume 42 / Issue 3 / 01 September 1999

Published online by Cambridge University Press:

20 November 2018, pp. 380-385

Print publication:

01 September 1999
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A lower bound on the number of points that can be placed in a square of side $\sigma$ such that no two points are within unit distance from each other is proven. The result is constructive, and the series of packings obtained contains many conjecturally optimal packings.

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