Folded forms

Manipulating geometries for fun and profit.

Folded Forms Lecture Series

Tucson Botanical Gardens

March 8, 2018

Shankar Venkataramani

School of Mathematical Sciences

College of Science

Origami: Endless potential

Flower power: NASA “Starshade” project

Credit

Koryo Miura and his famous fold.

Image courtesy Douglas Main

By MetaNest - Own work,

CC BY-SA 3.0,

Wikipedia

Lakshminarayanan Mahadevan

Lola England de Valpine Professor

of Applied Mathematics, Organismic

and Evolutionary Biology and

Origami saves lives: Stents

Top view of (a) inverted-cone fold imposed on an airbag in a housing, packed using the described

offset cross method for the inverted-cone fold pattern, (b) baseline fold imposed on an airbag in a

housing, folded using a traditional rectangular fold that has the corners compressed or forced

inward in order to fit within the circular perimeter of the airbag housing and (c) inverted-cone fold

imposed on an airbag in a housing, packed using the described nested cylinder method for the

inverted-cone fold pattern.

Jared T. Bruton et al. R. Soc. open sci.

2016;3:160429

© 2016 The Authors.

Origami in cosmology

Research and Images by

Mark Neyrinck et al,

The Origami Dynamics of the

Dark-Matter Sheet and Improved

Cosmological Constraints from

Understanding It

Mark Neyrinck

Nuala McCullagh

Alex Szalay

Johns Hopkins

University

Matter flows around with ~10 Mpc displacement from its initial comoving location. Around overdensities, BAO shells contract and get denser; around underdensities, BAO shells expand and get less dense.

This largely removes the BAO shift, making the location of the peak of ξA(r) more

faithful to its initial location.

↓ Fig. from Padmanabhan et al. (2012)

McCullagh, Neyrinck & Szalay (2013)

1D pos (2D pos-vel) 2D (4D)

In position-velocity phase space, the initial dark-matter sheet folds up like

origami to build structures.

3D (6D)

↑ Fig. from Kaehler, Hahn & Abel (2012)

Haloes, filaments,

walls and voids can be identified by the number of orthogonal axes along which their particles cross

com-pared to the initial conditions (Falck, Neyrinck & Szalay 2012) Like 2D flat paper origami, the graph of 3D dark-matter streams (polygons), bordered by

dark-matter caustics

The “crease pattern of the universe”

Be fo re f old in g A fte r f old in g

2D flat origami is 2-colorable: “up” and “down” colors

The usual mass-weighted correlation function

ξ_{δ(r) weights overdensities more than}

underdensities. Because gravity pulls overdense BAO shells toward their overdense centers, on average the BAO peak is shifted a bit inward.

(folds), is colorable with only 2 colors, without duplicating a color across a boundary (Neyrinck 2012).

In the usual matter power spectrum, power transfers from large to small scales (e.g. HKLM 1991, Peacock & Dodds 1996). This motion comes from over-weighting dense regions that contract. In the PDF-Gaussianized field, regions that grow less dense, and expand, are counted, too, so power moves both up and down in scale. This can be investigated by putting spikes at dif-ferent wavenumber k in the initial conditions, and seeing where the spikes go.

At left is shown the linear-power-propagation matrix

G

P

≡

Σ

G

P

i

Indeed, the spread of power is smaller and more symmetric in the Gaussianized field.

(Neyrinck, in prep)

Sim. details: 512 Mpc/h boxsize, 5123 _{particles, ALPT realizations (Kitaura & Hess 2013, see also Neyrinck 2013)} Try initial-density weighting instead of mass weighting.

For example, boost the weight of underdense regions with a “Gaussianizing” logarithm, δ→A≡log(1+δ).

Falck, Neyrinck & Szalay, 2012, ApJ, 754, 126, arXiv:1201.2353 Kaehler, Hahn & Abel, 2012, arXiv: 1208.3206

Hamilton et al., 1991, 374, L1

McCullagh, Neyrinck & Szalay, ApJL, 763, L14, arXiv: 1211.3130 Neyrinck 2012, MNRAS, 427, 94, arXiv: 1202.3364

Neyrinck 2013, MNRAS, 428, 141, arXiv: 1204.1326 Padmanabhan et al., 2012, MNRAS, 427, 2132 Peacock & Dodds, 1994, MNRAS, 280, L19

Download this poster/crease pattern, and other, simpler ones: http://skysrv.pha.jhu.edu/ ~neyrinck/origalaxies.html

References:

Origami follows rules!

Flat foldability

Not flat foldable!

Flat foldable

Intrinsic vs. Extrinsic geometry

Carl Friedrich Gauss

(1777-1855)

Princeps mathematicorum

(Latin

for "the foremost of mathematicians")

•

number theory and

algebra

•

statistics

•

analysis and differential equations

•

differential geometry

•

geodesy

•

geophysics

•

physics

•

astronomy

DISQUISITIONES GENERALES

CIRCA

SUPERFICIES CURVAS

AUCTORE

CAROLO FRIDERICO GAUSS

SOCIETATI REGIAE OBLATAE D. 8. OCTOB. 1827

COMMENTATIONES SOCIETATIS REGIAE SCIENTIARUM GOTTINGENSIS RECENTIORES. VOL. VI. GOTTINGAE MDCCCXXVIII

GOTTINGAE TYPIS DIETERICHIANIS

MDCCCXXVIII

Karl Friedrich Gauss

General Investigations

OF

Curved Surfaces

OF

1827 and 1825

TRANSLATED WITH NOTES

AND A

BIBLIOGRAPHY

BY

JAMES CADDALL MOREHEAD, A.M., M.S., and ADAM MILLER HILTEBEITEL, A.M.

J. S. K. FELLOWS IN MATHEMATICS IN PRINCETON UNIVERSITY

Gauss curvature

Theorema Egregium: The Gauss curvature relates/constrains

the extrinsic and the intrinsic geometries.

Theorema Egregium

Crochet courtesy Gabrielle Meyer

University of Wisconsin

Great Barrier Reef, near Cairns, Queensland, Australia.

Wikimedia/Toby Hudson,

CC BY-SA

The Crochet Coral Reef is a project

created by Margaret Wertheim and

Christine Wertheim of the

Institute For Figuring

Corals, crochet and the cosmos: how hyperbolic

geometry pervades the

universe

“These organisms are biological manifestations of what

we call

hyperbolic geometry, an alternative to

the

Euclidean geometry

we learn about in school that

involves lines, shapes and angles on a flat surface or

plane. In hyperbolic geometry the plane is not

necessarily so flat.”

“We have built a world of largely straight lines – the houses we live in, the skyscrapers

we work in and the streets we drive on our daily commutes. Yet outside our boxes,

nature teams with frilly, crenellated forms, from the fluted surfaces of lettuces and

fungi to the frilled skirts of sea slugs and the gorgeous undulations of corals.”

Margaret Wertheim

Vice-Chancellor’s Fellow in Science Communication,

University of Melbourne

https://theconversation.com/corals-crochet-and-the-cosmos-how-hyperbolic-geometry-pervades-the-universe-53382

Hyperbolic (non-Euclidean) Geometry

Sharon et al. 2004

Halftone Gel Lithography

Dynamic geometries

Hydrogel with

programmed geometry.

Eran Sharon et al,

Hebrew University.

Saddle shaped roofs

Catalina Church

Midtown Tucson

Monkey-saddles = hyperbolic

origami

Lines of inflection are analogs of creases in origami.

Scales of smoothness

Growing surfaces

Computation/images by Toby Shearman

My Ph.D students

John Gemmer

Toby Shearman

Curved Origami? But of course…

Flowers manipulating geometry: Time lapse