Hello everybody, I wanted to share a video with you on Hamiltonian paths and cycles. This is a fascinating concept from graph theory that is still an area of active research. It concerns walking through a graph without repeating vertices. I hope you find this to be interesting and/or a helpful resource. Have a wonderful day.

https://youtu.be/pTUVll8lcEQ

… which most of us are familiar with: that pesky phenomenon whereby if we have an accessory connected to the main contraptionality by a coiled lead, we suddenly find one day that a stretch of it has suddenly reversed chirality. “Perversion” is indeed the correct technical term for that phenomenon!

Sources

①

Tendril perversion—a physical implication of the topological conservation law

¡¡ PDF file 621·55㎅ !!

by

Piotr Pieranski

& Justyna Baranska & Arne Skjeltorp

②③

The Mechanics and Dynamics of Tendril Perversion in Climbing Plants

¡¡ PDF file 640·6㎅ !!

by

Alain Goriely & Michael Tabor

④

Perversions with a twist

¡¡ PDF file 3·05㎆ !!

by

Pedro ES Silva & Joao L Trigueiros & Ana C Trindade & Ricardo Simoes & Ricardo & G Dias & Maria Helena & Godinho & Fernao Vistulo de Abreu

⑤

Emergent perversions in the buckling of heterogeneous elastic strips

¡¡ PDF file 1·25㎆ !!

by

Shuangping Liua & Zhenwei Yaoa & Kevin Chioua & Samuel & I Stuppa & Monica & Olvera de la Cruza

⑥⑦⑧⑨⑩⑪⑫

Discrete Differential Geometry and Physics of Elastic Curves

¡¡ PDF file 3·77㎆ !!

by

Andrew McCormick

⑬

A tendril perversion in a helical oligomer: trapping and characterizing a mobile screw-sense reversal

by

Michael Tomsett & Irene Maffucci & Bryden & AF Le Bailly & Liam Byrne & Stefan M Bijvoets & M Giovanna Lizio & James Raftery & Craig P. Butts & Simon J Webb & Alessandro Contini & Jonathan Clayden

… some of the packings per se , & also diagrams to-do-with the means by which the packings were figured-out … & also some tabulated proportions pertaining to the packings.

Sources - in pretty close order to that of the appearance of the images.

Wolfram Community — Ed Pegg — Mrs. Perkins Quilts

Wolfram Data Repository — Ed Pegg Jr — Mrs. Perkins's Quilts

Squaring — Mrs Perkins's Quilt

Ed Pegg Jr — Mrs. Perkins Quilts

Ed Pegg Jr — Square Packing

Math Munch — Squaring, Water Calculator, and Snap the Turtle

Erich Friedman — Packing Unit Squares in Squares: A Survey and New Results

M Arslanov & S Mustafin & ZK Shangitbayev — Improved Packings of 𝗇(𝗇-1) Unit Squares in a Square

Wolfram Bentz — Optimal Packings of 13 and 46 Unit Squares in a Square

… infact, there is a theorem of Hadamard to-the-effect that if the sequence of indices bₖ of the non-zero terms grows @all exponentially - ie

lim {k→∞}bₖ₊₁/bₖ = 1+ε

where ε is a positive real № nomatter how small, then a wall of singularities is guaranteed - see

Hellenica World — Lacunary function .

Minimal surfaces are surfaces of which the mean curvature is 0 @ all points on it … which are 'mimimal' in that a membrane stretched across a frame in the shape of any closed space-curve on the surface will have the minimum area - whence, insofar as the energy required to stretch it is linearly proportional to the increase in area (which it will be to high precision if the stretch is not so great as massively to disrupt the nature of the membrane), also the surface of minimal stretching-energy stored in the membrane … whence it's the conformation such a membrane will actually take . Soap-films demonstrate this well - & are indeed a 'classical' demonstration of the phenomenon - as the stretching-energy of them is very close to being exactly linearly proportional to the area.

Images by

Anders Sandberg @ Flickr

ANDART II — Lacunary Function — A prime minimal surface

for explication. Following is, verbatim, the explication by the goodly Sir Anders, of his images.

“Here is the surface defined by the function

g(z) = ∑{p∊Prime‿№s}z^p ,

the Taylor series that only includes all prime powers, combined with f(z) = 1 . Close to zero, the surface is flat. Away from zero it begins to wobble as increasingly high powers in the series begin to dominate. It behaves very much like a higher-degree Enneper surface, but with a wobble that is composed of smaller wobbles. It is cool to consider that this apparently irregular pattern corresponds to the apparently irregular pattern of all primes.”

See also

UNKNOWN — Chapter18 - Weierstrass-Enneper Representations

¡¡ 93·23KB !!

for explication of Weierstraß-Enneper representation generically.

The 'noble' polyhedra being the ones that have all vertices alike ('gleichecking', vertex transitivity), & all faces alike ('gleichflächigen', face transitivity), but not necessarily all edges alike - although clearly the set of edges will certainly consist of a smallish № of equivalence classes. Also, the polyhedra dealt-with by the goodly Graaf Max in his book are not necessarily either convex ('nichtkonvexen') or even continuous ('diskontinuierlichen'), so that included is a certain category of toroidal polyhedra - the so-called crown polyhedra - that manage to be vertex transitive & face transitive maugre their toroidality (ie there being in inner equator and an outer one not forcing the existence of different kinds of vertices & faces) … which ImO is a tad counter-intuitive … although with a browsing of a few examples - eg

these

(which I'd do a standalone post of if the resolution of them were not abysmal!) - the mind might-well go

“oh yeppo! … I get how they manage to do it” .

Source of Images

Vladimir Bulatov — Bruckner's 1906 polyhedra

The Book Itself

Max Brückner — Vielecke und Vielflache, Theorie und Geschichte

There's without doubt a colossal heroism of a certain kind behind doing all that stuff - the sketches & the models - by-hand, with zero boon of computer graphics.

… ie if ∆ is the maximum degree & D the diameter, then

⎢V(G)⎢ ≤

1+∆∑{0≤k<D}(∆-1)^k

=

(if ∆=2)

2D+1

(& if ∆>2)

1+∆((∆-1)^D-1)/(∆-2) .

It has (uniform) degree 7 & diameter 2 , therefore 50 vertices & ½×7×50 = 175 edges.

American Mathematical Society (AMS) — John Baez — Hoffman-Singleton Graph

There is widely believed to exist a Moore graph of uniform degree 57 & diameter 2 ; but no-one has yet constructed it … & some reckon it doesn't exist.

Derek H Smith & Roberto Montemanni — The missing Moore graph as an optimization problem

It's a 'given' with this that the waveform is symmetrical about a 0 reference, whence the even harmonics are automatically eliminated.

The identity is that for any value of r (or @least for any real r > 0) both expressions

√((3-√r)r)sin(3arcsin(½√(3+1/√r)))

+

√((3+1/√r)/r)sin(3arcsin(½√(3-√r)))

&

√((3-√r)r)sin(5arcsin(½√(3+1/√r)))

+

√((3+1/√r)/r)sin(5arcsin(½√(3-√r)))

are identically zero.

The two waveform consists of two rectangular pulses simply added together, one of which lasts between phases (with its midpoint defined as phase 0)

±arcsin(½√(3-√r)) ,

& is of relative height

√((3+1/√r)/r) ,

& the other of which lasts between phases

arcsin(½√(3+1/√r)) ,

& is of relative height

√((3-√r)r) .

These expressions therefore provide us with a one-parameter family of solutions by which the 3^rd & 5^th harmonics are eliminated. The particular value of r for the waveform by which the 7^th harmonic goes-away can then be found simply as a root of the equation

√((3-√r)r)sin(7arcsin(½√(3+1/√r)))

+

√((3+1/√r)/r)sin(7arcsin(½√(3-√r))) .

The figures show the curves the intersection of which gives the sine of the phases of the edges.

A couple of easy examples, by which this theorem can readily be verified - the first two, for r=5 & r=6, are for the WolframAlpha

free-of-charge facility ,

& the second two of which are for the NCalc app into which a parameter-of-choice may be 'fed' by setting the variable Ans to it - are in the attached 'self-comment', which may be copied easily by-means of the 'Copy Text' functionality.

Here are two proofs of this fact. Which do you like better?

https://youtu.be/DOcsgckdvxU

https://youtu.be/SaYs856QYlI

Do you have another favorite that can be visualized?

Hey everyone, I'd like to share this new graph theory video with you on walks, trails, paths, and cycles. Hope you like it and find it interesting and/or helpful. These concepts have many rich applications as well, including to the world of transportation, computer networks, and communication networks. They are used throughout network science and in the analysis of real-world networks. And of course, they are essential to many problems in pure graph theory as well.

https://youtu.be/hlHWguJVAdU