#polyhedron net

[knotty-et-al]

Colorful nets of Truncated Tetrahedron and Tetrahedron

[art-of-mathematics]

Question for you: What kind of polyhedron will it fold into?

[dailypolyhedra]

Polyhedron of the Day #63: Triakis tetrahedron

The triakis tetrahedron is a Catalan solid. It has 12 faces, 18 edges, and 8 vertices. It is also known as the kistetrahedron, as it is the Kleetope of the tetrahedron (i.e., this shape can be obtained by attaching triangular pyramids to each face of a tetrahedron). Its structure is similar to that of the net of the four-dimensional 5-cell. Its faces are all isosceles triangles. Its dual polyhedron is the truncated tetrahedron.

Triakis tetrahedron GIF and image created by Cyp using POV-Ray, distributed under a CC BY-SA 3.0 license.

[shmowder]

Nothing could convince me that Daniil and a psychologist reader wouldn’t psychoanalyze eachother as a way of flirting nothing more romantic then listing off all the personality disorder symptoms your partner has over a candle lit dinner

But yeah being a psychologist stuck in the town during the events of the game(s) would be utter hell that would probably cause some form of sanity slippage after some point there’s neurosis everywhere in that town and sure maybe one fire is put out but oops there’s four more in its place so after awhile it’s gonna cause them to bang their head against a desk or two….maybe even a decent wall if one can’t find a proper desk!

I think a psychologist would really shine through and become crucial in the aftermath of the plague when the town is attempting to heal itself.

Whether you're with the utopians utop the tower or down behind the wall of soldiers watching the polyhedron comes crashing down, your importance becomes very apparent very fast.

The Stamatins who come to you mourning their tower, Aspity and Oyun struggling to come to terms with the death of mother earth, Maria who suddenly can't hear her mother's voice anymore, Stakh who's lost all direction in his life overnight.

You'd be more than encouraged to stay. Your presence would be demanded if anything. As the last plague cloud dissipates from the streets, the wounds are still too fresh in everyone's hearts.

And while the doctors in town can tell you a lot about how to stich cuts, heal broken bones, and cure illnesses, they can't help mend scarred minds.

A psychologist would be the safety net catching the afterfall of those two nightmarish two weeks.

[study-with-aura]

Monday, February 5, 2024

It is good to be back and posting again after my semi-weekend break. I type semi- because I did post in response to a few messages. However, I did take a break from my intensive studying this weekend, and only kept up with reading and Duolingo.

I do hope everyone has a lovely week and practices self-care daily and when needed throughout the day.

Tasks Completed:

Geometry - Learned about polyhedrons + identified parts of solids + identified solids + identified nets of solids + practice

Lit and Comp II - Copied Unit 17 vocabulary + read chapter 11 of Emma by Jane Austen + read "A Hunger Artist" by Franz Kafka + completed an online story diagram + created an antagonist and conflict for the short story I'm writing (the library has their flash fiction competition coming up, so I'll use this story for that more than likely)

Spanish 2 - Listened to a story in Spanish + reviewed vocabulary

Bible I - Read Deuteronomy 29-30

World History - Wrote an essay explaining how the rise of the Young Turks, the Russo-Japanese War, and the Boxer Rebellion are tied to nationalism

Biology with Lab - Made food chains with an online simulation + continued doing research on my endangered animal (saola)

Foundations - Read more on patience + completed Lumosity daily brain workout + read about special occasion speeches + defined different types of special occasion speeches + chose a type of special occasion speech for my speech assignment (I went with inspirational, and I want to do an inspirational speech for the younger girls in our Girl Scout troop)

Piano - Practiced for one hour

Khan Academy - None today (already assigned)

CLEP - Completed Module 8.5-8.6 lecture videos

Duolingo - Completed at least one lesson each in Spanish, French, and Chinese

Reading - Read pages 356-401 of House of Marionne by J. Elle and finished the book

Chores - Cleaned windows in my bedroom and in the study + took the trash and recycling out

Activities of the Day:

Volunteer for 2 hours at the library

Ballet

Contemporary

Journal/Mindfulness

-

What I’m Grateful for Today:

I am grateful for the opportunity to give back to my community.

Quote of the Day:

Hope can be bruised and battered. It can be forced underground and even rendered unconscious, but hope cannot be killed.

-UnSouled, Neal Shusterman

🎧Romance, Op.1/4 - Ossip Gabrilowitsch

[lady-inkyrius]

@kaiasky

Yeah, tetrahedral maps are great! I think it's mainly because when the map is centred on the north pole you sort of get three "prongs" of land extending from the centre (South America, Africa, and Australasia/Oceania), which then obviously fit into the three points of the Lee Tetrahedron's resulting triangle really well.

This manages to keep most of the land away from the distortion points at the centre of each side and at the corners, with the noticeable exceptions of the area around Myanmar & Bangladesh, Hawaii, and Antarctica.

I'll put the rest of this under a cut because it got quite long.

There are a couple of different ways to fix this, centring it on 65°N 30°E rather than the pole manages to put Antarctica back in one piece and now the only noticeable distortion is that Sri Lanka and southern India are way too big, (There are probably better options than this for the centre but I couldn't find them)

Of course, you might also notice that the three "prongs" are really closer to 90° apart than 120°, which is where the popularity of octahedral projections like Cahill's and Waterman's projections probably comes from, but I quite like the Peirce Quincuncial for this, and if you take the top octant and split and rotate it round, you get a layout very similar to the tetrahedron. The most noticeable distortion here is probably Papua New Guinea being slightly to big.

(And like in this post from last week, you can also move the bit of Antarctica at the bottom to get this)

A lot of people have tried to use Lee's Tetrahedron as a base and either rearrange it into a rectangle (Markley and CALM) or a half hexagon (Lee-Concialdi and Lee-Xarax), it's just a really good projection if you need something conformal with low area distortion.

The only oblique aspect rearrangement of the Peirce Quincuncial I've seen is Grieger's Triptychal (see the Markley and CALM link).

A polyhedral projection I really want to see is one based off a rhombic dodecahedron, a polyhedron with twelve rhombi for sides.

It's net isn't particularly pretty because the rhombi it uses don't tesselate, however this paper does a really interesting thing by compressing the net to squares rather than rhombi.

It's original use case is as replacement for cubemaps which can cause a loss of visual quality where the distortion is large, but I want to see what it would look like for a world map.

The projection they're using for each face (gnomonic I think) isn't conformal so you get discontinuities between face, but you might be able to change it so it is, sort of like how the Peirce Quincuncial uses right angle triangles for each octant rather than equilateral triangles like the Cahill while still maintaining conformality, but I don't know how I'd work out the maths for that, iirc Peirce did it with complex analysis.

[lumeha]

🧡🤍?

🧡: What is a popular (serious) theory you disagree with?

Is it still popular ? I Don't Know, BUT

Patho 1 Clara's ending being a net positive because both the Town and the Polyhedron can survive together at last, at the price of blood.

Because for me it's genuinely the "Oops Now You've Got Blood On Your Hands Girl !!" ending. The "Well You've Done It, You've Fucked Up Something, And You Have To Believe It's Good Because You Can't Go Back" ending. Which.

In a way, kind of all the endings have shades of it, if I can say it that way, but idk, I feel like especially with Clara, it's even more of a thing. She's the one who knows it's all a game. She's the one who goes all about not having blood on her hands. What does it say of her when she... doesn't get the super special dev ending for the player ? What does it say that she convinces herself that human sacrifice is the answer ??? That because it's in her hands it's okay, because it's not true, or is it true ????

... is this even a theory, or is that an interpretation ? Well. Hm. Fuck that I guess, but you're getting that because I typed it.

...... worst part being that it's been a while since I touched Patho 1 and I probably should touch it again before talking about it, especially about Clara, but (shrugs) is anyone going to really get mad at me for this, it's not even my worst Patho take

🤍: Which character is not as morally bad as everyone else seems to think?

(mumbles) I'm trying to stop answering Rhea stop asking me questions that would make me answer Rhea >:( (joking)

... You know what that's genuinely an excellent question and I somehow cannot find it in myself to answer anything but Rhea. I've been searching for like 10 minutes in my brain, but I'm just blanking out. Most antagonists that I find sympathetic and not morally awful are *usually* kind of considered like that by the fandoms as a whole, and for main and secondary characters, I'm just

I'm drawing a blank, and the blank is in the shape of the dragon pope, because people love to villainize her for no goddamn reason and I'm so tired of it ??

(fun fact I was even trying to think abut YGO but I guess I'm also not involved enough to really pick who is considered morally bad and I would disagree on) (the advantage of being disconnected from fandom ?)

[paulfc]

Pull up nets, cube, tetrahedron, octahedron, dodecahedron and icosahedron

I was reading a paper about pull up nets and it suggested that not all of the nets of a cube necessarily can be made to pull up.

Challenge accepted, I have made all eleven of the cube nets and depending on your definition of 'pull up' all of them can be made to pull up.  I accept that not all of them do so elegantly but they do pull up into a cube.  Given that there are many ways to loop thread through them improvements may be possible if there were an easy way to determine the most effective route.

When considering this, knowing which bit of the net end up where in the cube, that is to say which corners in the net end up in which vertices in the cube helps in choosing a thread route that pulls these together.

Tacked on to the end are the only two unique nets of the tetrahedron, one of the eleven nets of the octahedron and one each of the thousands of nets of the dodecahedron and the icosahedron.  I may get round to the other ten nets of the octahedron but not the thousands of the dodeca and icosahedrons.  The number of nets increases geometrically with the number of faces and side of the polyhedron I don’t know if there is a formula but it does get very silly very quickly.

These are card with sticky tape hinges and the eagle eyed will have seen that sometimes they flex the wrong way and yes the card came from pizza boxes, I eat a lot of pizza and always save the cardboard for making maquettes.

[kaishirase]

Stella Users' Polyhedron Models

Fr. Magnus Wenninger | Ulrich Mikloweit | Tom Lechner | Piotr Pawlikowski | Kyle French | Sergey Kaliberda | Sebastián Naccas | Hamp Stevens | David Bodoh | Alex Doskey | Adam Stolicki | Giacomo Artoni | Richard Stratton | Marc Picquendar | Robert Rech | Linda Zurich | Karlos Alonso Mediavilla | Steve Waterman | Michael Barltrop | Keith Davison |

Here are some models made by people other than me, using measurements or nets generated by Stella4D, Great Stella or Small Stella.

If you have made models using Stella, please email me (Stella4D@gmail.com) some images, and I may include them here.

See also: What people have to say about Stella.

Models by Fr. Magnus Wenninger

Author of "Polyhedron Models", "Dual Models" and "Spherical Models"

[knotty-et-al]

I drew that rhombicuboctahedron from a cuboctahedron.

I also collected all polyhedron drawings from this notebook in an own file - and started to add infos such as number of vertices, edges and faces to each page. I will also add further infos and stick them on the pages once I know how I want the infos to be placed and which infos I furtherly want to add.

I want to add polyhedron nets as well.

[art-of-mathematics]

Icosahedron net - slightly colorful visualization

Currently I am obsessed with polyhedra nets - and crafting tiny models and creating visualizations.

For this one I created a template for the net that somehow reminds me of the cornu spiral - as this kind of net can be "coiled up" to fold the polyhedron.

Furthermore, I used 12 colors for the 12 vertices - in a gradient-like arrangement (it starts at red and gradually follows the route at orange, yellow, green, blue, violet, and ends at pink.)

[dailypolyhedra]

Polyhedron of the Day #14: Cube

The cube, also called a regular hexahedron, is a Platonic solid. It has 6 faces, 12 edges, and 8 vertices. It can also be seen as a square parallelpiped, equilateral cuboid, right rhombohedron, 3-zonohedron, or regular square prism. It is also a Hanner polytope. The dual of the cube is the octahedron. It is the only convex polyhedron whose faces are all squares. Its primary Schläfli symbol is {3,4}. The cube has three uniform colourings and eleven nets.

Cube GIF and image created by Cyp, distributed under a CC BY-SA 3.0 license.

[spring-of-mathematics]

I built some of my polyhedra

a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1)

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids which are not uniform (i.e., not a Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 Johnson solids (excluding the 5 Platonic solids, the 13 Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms), and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (J2) is an example that has a degree-5 vertex.

A database of solids and polyhedron vertex nets of these solids is maintained on the Sandia National Laboratories Netlib server (https://netlib.sandia.gov/polyhedra/), but a few errors exist in several entries. Corrected versions are implemented in the Wolfram Language via PolyhedronData.

The following list summarizes the names of the Johnson solids and gives their images and nets.

[wirsindkrieg]

The Lens of Human Perception

There’s a topic I’ve realized I haven’t discussed here, so I figure it’s time to write something about it. As usual, due to the length of this essay, the bulk of it is under the cut.

I’ve seen a lot of posts going around by people exploring nonhuman past lives, especially those which aren’t possible here/now (i.e., angels, dragons, etc.), and describing any memories they have in very human terms. Things like walking down streets, gathering in gardens or libraries, standing before a judge in court during a trial. I think this is the result of a lack of awareness of a phenomenon we all deal with, but which is rarely (if ever) discussed directly.

Every memory you have, from the broadest strokes to the finest detail, is filtered through the perception and understanding we have here/now. We’re experiencing three-dimensional space, with a (relatively) constant and linear flow of time, seeing the world by way of reflected light and hearing things by way of vibrations in the air around us. Unfortunately, because those things are the only way we’ve experienced reality in this lifetime, we are literally incapable of properly modeling other forms of reality in our minds. It’s a limitation of being here/now, and having the kinds of experiences we do.

A (slightly extended) example: It’s pretty common to be taught about nets of polyhedra in school these days. As a refresher for those who haven’t dealt with the concept in a while: Polyhedra (singular “polyhedron”) is a three-dimensional figure made up of flat planes joined at some angle. One of the most common polyhedra is the cube. Six flat faces, joined at right angles. A net of a polyhedron is a way of “unfolding” the faces of a polyhedron into a two-dimensional shape. That two-dimensional shape can then be folded through the third dimension to recreate the original polyhedron.

Nets aren’t limited to unfolding three-dimensional objects, however. Theoretically, you can create a one dimensional net of a two dimensional polygon (it’ll just be a line, but it counts). More significantly for this discussion, though, you can create three-dimensional nets of four-dimensional shapes. Some of you may be familiar with the concept of a hypercube (sometimes called a tesseract). In the same way that a cube can be thought of as being made of squares arranged in three dimensions, a hypercube can be thought of as being made of three-dimensional cubes arranged in four dimensions. (There are multiple ways of representing a hypercube in three dimensions, but all of them are by necessity inaccurate.) Now, I’d like you to take a moment, and try to imagine a proper four-dimensional hypercube, and how it unfolds into a three-dimensional net. Then try to imagine that three-dimensional net folding back into a four-dimensional hypercube.

It is, quite simply, impossible to do accurately. Our brains are simply incapable of truly visualizing four-dimensional space, and the way objects behave in four dimensions, because we’re limited to experiencing only three of them. Similarly, it is impossible to truly comprehend how reality functions outside of here/now, because we’re limited to experiencing only here/now. Brains are amazing things, however, and will try their hardest to fill in the gaps with something, even if that something is inaccurate.

When it comes to remembering nonhuman incarnations, this means that our brains will grab whatever feels closest to what you’re unable to understand, and use that to fill in the gap, regardless of if it actually fits or not. As a result, those kinds of memories are inherently unreliable, since they’re (in many ways) just our brain’s best guess about what actually happened.

I’ve taken to referring to this problem as “the lens of human perception”, since it occurs due to filtering things through the understanding we can have by perceiving the world the way humans do. Like any imperfect lens, it leads to our view of things being distorted, and because of the way brains work, we generally aren’t even aware that the distortion has occurred. And so we redefine things in terms of here/now without even realizing we’re redefining our own experiences.

Being aware of the lens of human perception can be helpful, however. It lets you consider the context of your memories, and figure out where the gaps in your understanding are getting filled in. You can then make an effort to explore those gaps, and refine your understanding to get a clearer picture of how things actually were. It’s difficult sometimes, and takes some work to do effectively, but it can be worlds better than getting caught up in misunderstandings about your own experiences.

So if you’re exploring memories and notice that things are very similar to the world around you, take some time and question things. Think about what it is that you might not be able to understand within those memories. Reflect on what unconscious associations you might have with the here/now things your brain is using to fill in the gap. Use those associations as a starting point to explore further, and refine your understanding into something more accurate to what you really experienced. You may not be able to remove the distortion from the lens completely, but you can at least compensate for it. With time and effort, it’ll help you be more certain about your experiences, and know that you have a clearer view of yourself. And believe me, it’s a great feeling when you get there.

[kitwallace]

Paths on Polyhedra

Bob Bosch posted a picture of a rather lovely wooden object.  A grove had been machined around a cube in which a steel ball was free to run.  

Ever willing to steal other folk’s great ideas, I set about implementing the object in openSCAD.  My first attempt was a hack.  I made a quarter-circle tube out of hulled spheres and translated and rotated copies to lie with the right orientation on the sides of a cube. The tubes were then removed from the cube using difference(), leaving a continuous grove. Displacing the tube inwards created a lip to capture the ball bearing.

It was only when I had made the construction that I realised that the path was closed and touched every face so it was a Hamiltonian circuit around the cube faces. [I noted later that it was the waist of the cube too]

A better construction would take a list of face indexes and convert to the sequence of quarter tubes.  Given the definition of the polyhedron as a set of faces and vertices, it should be possible to construct the tubes in their position on the surface. 

It helps to number the faces:

In openSCAD you can rotate this object to find a cycle of  [0,1,4,3,5,2] .  Each tube runs across a face from mid-point of one edge to mid-point of the next. 

The construction interpolates along the arc to to create a path as a sequence of points, then places spheres on the path which are hulled together. Now any path can be constructed, like the loops around the bare vertices:

This code is generalised so we can construct paths on other polyhedra:

Tetrahedron:

Octahedron: 

When edges are adjacent, the path is an arc.  If they are not, the path is a straight line between the edge midpoints.  This is also a circuit, but less symmetrical :

Performance is slow and an alternative construction using a constructed polyhedral net would be faster and allow a tube profile which could also round the sharp edge.

It would be fun to commission one of these objects - with a captive ball - in metal from Shapeways.  I fancy making a dodecahedral version, using the vertex Hamiltonian of its dual icosahedron. I think its rather curious that Hamilton’s Icosian game was played on the vertices of a dodecahedron, rather than on the faces of a icosahedron, which would seem to be easier to draw. 

 Code on Github.  

[notreallyherehahaha]

A Symmetrohedron Featuring Regular Heptagons, Regular Hexagons, and Irregular Pentagon-Pairs

A Symmetrohedron Featuring Regular Heptagons, Regular Hexagons, and Irregular Pentagon-Pairs

Symmetrohedra are symmetric polyhedra which have regular polygons for most (but not necessarily all) of their faces. I made this particular one using Stella 4d, which you can try for yourself at this website. Here’s the net for this polyhedron, also.

This particular symmetrohedron features 12 faces which are regular heptagons, and 14 faces which are regular hexagons. The irregular faces are 24…