#polar graph plots

[xyymath]

Polar Graphs: When Math Decided to Flex Its Artistic Side

The Polar Rose (r = asin(nθ))

2. The Lemniscate (r² = a²cos(2θ)) : It’s like infinity had a baby with a pretzel

3. The Cardioid : r = a(1 + cos(θ))

4. The Spiral of Archimedes (r = aθ)

5. The Butterfly Curve :

[azurepolarbear]

💚🤍

The sine function is a trigonometric function whose output oscillates between one and negative one. Plotting the function in two-dimensional space results in a wave-like graph, whose appearance can change based on the length, amplitude, and frequency of the wave.

“rainbow waves v2.0” is an original generative code art algorithm; each run of the code produces a random visual output. Sine waves are created and stacked on top of each other. Each wave is made from a random number of points, with a random amplitude and frequency. Points are assigned their colors from a color selector that chooses colors from a random palette.

“rainbow waves v2.0” was made with TypeScript, p5.js, and the @batpb/genart library.

The source code of this project is licensed under the GNU Affero General Public Version 3.0 License.

The visual output of this project is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) License.

Copyright (C) 2015-2024 brittni and the polar bear LLC. Some rights reserved.

[gamingavickreyauction]

The shape of a y=x^2 graph isn't maths. There's no way to prove what it looks like.

Think about when you first learn how to draw a graph: you do it by finding various points on the graph and plotting them, then joining the dots. This plotting is the key to drawing a graph: plotting is a map from abstract mathematical points—really just symbols—to real points, in physical space. And we all learn how to do it early. But there's nothing more right about how we do it than any other way. This might seem obvious—after all, you can plot a polar graph too—that's just a graph from another different way of plotting—there are any number of ways you can parameterize the plane.

One could imagine a completely alien way of plotting points on a page, where straight lines become eldritch monstrosities. This is no more right or wrong than our way of plotting—neither have any justification mathematically—the process of plotting can't even be described mathematically.

One could plot a ZFCircle in this system, and all its properties would translate into properties of the eldritch circle, in just the same way as they do for our circles. ZFCircles don't 'look like' our circles any more than they 'look like' those eldritch circles. All the reasons we say ZFCircles correspond to our circles are also reasons ZFCircles correspond to eldritch circles. Both are equally valid ways of translating ZFCircles to physical space. So I don't see how ZFCircles can be said to capture any notion of shape at all.

You might think we can describe the plotting mathematically: we can model the paper we draw on with R^2 and, for example, plotting as a radial graph is then described by the function: (r, θ) → (r cos θ, r sin θ), and the alien plot is described by the ugliest formula imaginable. The Cartesian plot is described by the identity map. But that's only the case because our starting point is an analogy between R^2 and our graph paper, using the Cartesian plot. If, for example, we thought of R^2 in radial coordinates, we would describe the Cartesian plot as (x, y) → (sqrt(x^2+y^2), arg(x, y)).

So we describe the same plot with a different function depending on which other plot we are used to using. Which means we cannot actually capture the plot itself in maths—only how it relates to other plots—which we capture as the notion of a change of coordinates.

But then, analogizing a plot to a function can't break us out of the purely algebraic world and let us describe shapes. That function that we tried to take as an analogy for the radial plot isn't any more objectively an analogy for the radial plot than any other bijection R^2 → R^2. So studying this function cannot directly tell us anything about the radial plot.

The only way we get information back about the radial plot is by translating algebraic facts about that function into geometric facts using the Cartesian plot we started with—which we were trying to justify in the first place. Attempts to justify the analogy between shapes and algebra mathematically are doomed to always end up just chasing their own tail.

Geometry is founded on faith.

[dracomaton-integration]

hey don't cry, plot 1/5-1/m = x/m on a polar graph where x and m are both whole numbers and m is the modulus of the circle for a chance to see the Fivefold Tartan Dimension

2/5-1/n also works

[ansibytecode]

Exploring the Techniques of Data Visualization

Large data sets need in-depth analytics and processing power to manage. This is where data visualization is helpful. Data visualization services have advanced rapidly in recent years, anticipated to alter the business environment shortly.

Data visualization utilizes visual elements such as charts, graphs, and maps to facilitate the observation and comprehension of trends, outliers, and patterns in data. It helps determine which variables to include or discard in the analysis.

This blog on data visualization techniques will provide detailed insights into the techniques and benefits.

What is Data Visualization?

Data visualization is a captivating form of visual art that captures our attention and effectively conveys a message. When we look at a chart, we can easily identify trends and outliers. Visualizing data allows us to quickly internalize information. It's essentially storytelling with a purpose. If you've ever struggled to identify a trend in a large spreadsheet of data, you understand the power of visualization.

Data visualization is a powerful method to explore data and present results effectively. Its primary use is in the pre-processing stage of the data mining process. It supports the data-cleaning process by identifying incorrect and missing values.

Techniques of Data Visualization

Representing visual data requires various techniques that must be followed to achieve this process. Let's explore some of these techniques to make this process simpler and easier.

1. Temporal

Temporal data visualization offers the advantage of familiarity, as we are already accustomed to using such visuals, particularly in educational and professional settings where charts are commonly used for explanations. Linear and one-dimensional data visualizations play a crucial role in temporal data visualization. Examples of temporal data visualizations include linear graphs, polar area diagrams, scatter plots, time series sequences, and timelines.

2. Multidimensional

Multidimensional data visualizations, as their name implies, involve multiple dimensions, typically requiring at least two variables for a 3D data visualization. These visualizations often feature vibrant and striking graphics due to the numerous concurrent layers and datasets. They excel at condensing large amounts of information into key points. Examples of multidimensional data visualizations include histograms, scatter plots, pie charts, Venn diagrams, and stacked bar graphs.

3. Geospatial

Geospatial or spatial data visualizations involve overlaying different data points on familiar maps to connect them to specific geographic locations. Examples of geospatial data visualizations include Cartograms, Heat maps, Flow maps, and Density maps.

4. Network

Users of network data visualization can demonstrate connections between different data sets. Within this network, communication takes place via intricate connections linking one data set to another. Visualizations such as alluvial diagram charts, parallel coordinate plots, node-link diagram charts, word cloud plots, network diagram charts, non-ribbon chord diagram plots, and matrix charts are commonly used to illustrate the relationships between data sets.

5. Hierarchical

When information needs to be organized into clusters, hierarchical data visualizations are very helpful. However, creating these graphs is more complex compared to other forms of visualization. Hierarchical data visualizations can show a company's or organization's data and object hierarchy. Examples of hierarchical data visualizations include ring charts, sunburst diagrams, and tree diagrams.

Wrapping Up

Data visualization solutions are an essential step in data processing techniques. In the new era, data visualization is making its debut. With the introduction of next-generation technologies and the development of apparent frameworks, it is moving from art to science, opening up new opportunities.

Using the above guide, you can use data visualizations for processing your business data or use the help of data visualization consulting services. A leading data visualization company excels at this situation and can help you to implement this approach. There is a plethora of legacy modernization services available to modernize your business applications. To make data-driven decisions, choose top-quality data visualization services to create a data visual model.

[spring24mathdesign]

Project Two: "Solar Eclipse"

Project Two introduced Polar Functions! The designs created by polar functions were what I envisioned I would be making in this class (think Spirograph toys!)

Similarly to the first project, I didn't have much of an idea going into it, but once the pieces started coming together I found the inspiration to call my design "Solar Eclipse":

Most of the design was created by a stroke of luck and plenty of trial-and-error exploration. For most of the Polar Unit, my practice designs had the same outward burst effect. There was just something about focusing in on one aspect of the design and then branching out that appealed to me.

The Inspiration: The Solar Eclipse on April 8th, 2024

In truth, I did not use the eclipse as inspiration until later in the process. I originally wanted to create a sunrise/sunset scene because I had luckily found a curve that looked like clouds while trying out different function combinations. It was only after things started coming together in the design that I decided to base it off of the upcoming (at the time) solar eclipse.

Some Issues...

I had three main problems while working on Project Two.

The first of these was with setting a domain for the polar functions. Even though I had learned about the radians, I often found myself guessing-and-checking by putting random numbers next to pi and seeing if the graph was restricted the way that I wanted it to be.

Secondly, the markers that I used. Posca pens are known for their vibrancy, so I was a little disappointed that the color of the white marker came out watery. It took me more than a few times plotting the same inner ring for the marker to reach a satisfiable opaqueness which could have also put the paper at risk of tearing.

The third problem isn't one that I could have taken into my own hands. I don't know how the MakerSpace works outside of their machines and all my AxiDraw reservation times were taken up by someone else before I got to the space. I am regardless grateful that I was still able to copy the drawing twice despite the trouble.

Overall, I am happy with my outcome for Project Two! I can clearly see my growth between the first project and this one. I was still trying to get comfortable about borrowing cool-looking functions from other people at the time, but I think, even without taking ideas from my peers, this design still came out wonderful!

[sciencestyled]

When Paintbrushes Meet Test Tubes: A Comically Genius Guide to the Art-Science Mashup

Ah, the age-old debate: science versus art. It's like choosing between pizza and ice cream for dinner – both fantastic, but oh, so different. But what happens when you mash them together? You get a sundae topped with pepperoni, my friends – weird, but strangely satisfying. This is the space where beakers and brushes coexist in quirky harmony, creating a fusion of knowledge and beauty. And, of course, we're going to explore this delightful conundrum with as many pop culture references as humanly possible. Hold onto your lab coats and berets, folks!

First off, let's address the neon elephant in the room: Art and science are often seen as polar opposites. Science is all logic, numbers, and facts – the Spock of our story. Art, on the other hand, is the Captain Kirk – impulsive, emotional, and wildly creative. But, as any "Star Trek" fan will tell you, Spock and Kirk are better together, and the same goes for art and science.

Now, imagine if Leonardo da Vinci had decided to stick to just art or just anatomy. The world would have been robbed of a man who could sketch a Vitruvian Man in one hand and dissect the mysteries of the human body in the other. Da Vinci was the OG (Original Genius) of combining art and science. He didn't just paint pretty pictures; he used his art to dig into the scientific wonders of the world.

Fast forward a few centuries to the era of Instagram and TikTok, where art promotes science education in the most dazzling ways. Ever seen those hypnotizing videos where someone pours colored liquids into a petri dish, and it blossoms into a psychedelic display? That's art teaching science, folks! It's like watching "Breaking Bad," but instead of cooking meth, they're making art and teaching chemistry.

Speaking of chemistry, let's talk about the explosive reactions that happen when art and science collide. For instance, have you ever seen a sculpture that moves with the wind or changes its appearance with the angle of sunlight? That's not just art; that's physics and engineering donning a beret and calling itself "sculpture." It's like Iron Man building his suit – a perfect blend of tech and aesthetics.

And let's not forget biology, the science of life, which has been inspiring artists since the first caveman drew a woolly mammoth on his living room wall. Today, we have artists using living cells to create bioluminescent art. It's like "Avatar," but in a petri dish.

On the flip side, science benefits immensely from art. Ever heard of data visualization? It's the art of turning rows of snooze-worthy numbers into stunning graphs and charts that even a six-year-old could understand. It's like taking the plot of "Inception" and turning it into a Dr. Seuss book.

Moreover, art challenges scientists to think outside the proverbial box. A scientist might see a blob of cells under a microscope, but an artist sees a potential masterpiece. It's a bit like how MacGyver looks at a paperclip and sees a tool to save the world. Artists can inspire scientists to view their work through a different lens, adding a splash of creativity to their logical minds.

Now, let's talk about technology, the love child of art and science. From stunning video games that teach physics (Portal, anyone?) to virtual reality experiences that let you walk through a human heart, technology is the bridge between art and science. It's like if Tony Stark and Bob Ross had a baby – it would create beautiful landscapes with a suit of armor.

In conclusion, the intersection of art and science is like a Marvel movie – full of unexpected crossovers, collaborations, and a fusion of different worlds. It’s a place where creativity meets logic, where beauty intertwines with facts, and where education becomes fun. So, the next time you think about art and science, remember they're not just two separate subjects; they're two sides of the same coin, flipping endlessly in the air, waiting for us to catch them and marvel at their combined beauty. And who knows, maybe one day we'll have a school subject called "SciArt" – where students paint with chemicals and calculate the geometry of sculptures. Until then, keep mixing those test tubes with your paintbrushes and create some explosively creative magic!

[itonlinetraininginusa]

What are the core skills of a business analyst?

From an abstract standpoint, business analyst skills are a blend of hard skills and soft skills. This is done to draw attention to the fact that these skills have more to do with a person's emotional intelligence than intellectual ability. Intelligence is, of course, a component in determining their level of professional success.

Unlike solely technical skills, those needed for business analysts are not only learned via formal education. These skills are acquired through experience and a natural ability to read people and understand the context. Enroll in free business analyst courses to develop critical thinking and get employment opportunities.

Decision-making

It is expected of a business analyst to offer solutions that promote organizational expansion. The conclusions of a business analyst can significantly affect how an organization makes decisions. Making decisions based on a business analyst's findings can help your company outperform competitors and reach new heights.

Business analysts should consider all options carefully before concluding. Business analyst online training can help you make better decisions and adopt a mentality that considers multiple factors and foresees potential outcomes.

Skills to conduct stakeholders meetings 

Until now, using email to connect with a client efficiently and professionally has been the norm. It is not always the most efficient method, though. Therefore, talking about problems with the client face-to-face can be very effective and even speed up the resolution of problems. One of the key talents needed for business analysts is the ability to plan and lead meetings.

In fact, CEOs frequently have a greater understanding of a project or an issue because employees are more forthcoming in their contact with one another. But to schedule meetings for audit trials, written communication through email is efficient for business analysts. 

Data Visualization

Transforming unprocessed data into digital representations that can be used for business requires visualizing data. Business analysts with data visualization skills can help clients understand the data and make strategic decisions to satisfy their demands.

Business analysts employ many kinds of data visualization techniques, such as scatter plots, time series patterns, polar area illustrations, timelines, graphs of lines, and more. A business analyst online training course will make you skilled enough to use all these methods to create elegant and thorough visualizations.

Problem-solving

Having problem-solving skills makes business analysts successful in their careers. Every stage of the job of a business analyst is difficult, and they must be able to overcome these difficulties. Finding the right remedies requires careful consideration of the causes and sources of an issue. People who receive certified business analysis professional training online can efficiently solve problems in a company.

Ability to Understand Delegated Objectives

An important ability for business analysts is interpretation. It is important to understand the big picture and each requirement that the management or technical employees explain. There are often many gaps in the presented information, which the analyst must comprehend and fill in. One should not be reluctant to ask again if there is any confusion regarding the objectives. One of the most important business analyst abilities is the capacity for quick understanding.

Documenting and Writing Reports

Documentation is a great example of the technical skills needed for business analysts. It involves producing various analysis details, plans, reports, and documentation. A business analyst's duties will demand them to write about various topics.

Knowing when to use technical jargon and when to use plain, simple English is one of the most important aspects of producing reports effectively. This skill combines writing prowess and the capacity to understand communication constraints like the target audience and the message you're attempting to convey. Written communication is one of the most important skills taught in a business analyst online training course.

Final thoughts

In short, hard and soft skills combine to make up business analyst capabilities. Work on your interpersonal qualities and hone your technical skills if you want to become a good business analyst. Advanced business analysis certifications, in addition to the usual educational requirements and work experience, are helpful in landing employment with high incomes.

[mathblogtingz]

🗿pinakahuling math blog sa pisay #2🗿

This is my pangalawang blog tapos yung topic na is Polar Equations and Graphs. DI KO NAGAWA QUIZZZ . Madali lang sana yung quiz pero di ko nagawa.

In summary oks lang kasi plotting lang naman siya na may unting equations. Nakakainis lang talaga kasi di ko nagawa quiz nalimutan ko kasi tapos nangyari pa kasi steam week. Yun lang for this blog.

[jamdoingmath]

In which we enter the final revival of this blog.

How would you describe your math 6 learning journey?

-math 6 has been a rollercoaster throughout the years, however this quarter things have relatively settled down, from complex derivatives and integrals we enter a new dimension, (literally, the 3rd dimension!) we learn new concepts such as polar coordinates as well as adding a new axis to our regular cartesian graph, bringing us into the 3rd dimension as we learn distance formula all over again. while the concepts are new it brings a refreshing new scene into our math 6 journey.

Which topics did you enjoy the most?

Probably the introduction to polar coordinates, and even to 3d coordinates. Polar plotting and graphing is exciting to see the shapes take form on a new and interesting graphing system.

Like seen in this cheat sheet for polar graphs. The shapes formed by equations can be very visually striking.

What concepts did you find easy to learn?

-3d coordinates since i have had some experience with the 3d coordinate planes through video games and as well as 3d modelling programs like blender it was fairly easier to grasp, that although the classes alloted for the concept were few it was something i grasped quite quickly.

What concepts did you find most interesting?

-It would still be on 3d graphing and plotting. Since its something that i can apply more tangibly in my day to day life especially in my hobbies!

what concepts have you mastered the most?

polar graphing probably! It was something quite new but i had to learn properly, its quite tricky but also fun!

what concepts have you mastered the least?

Polar to cartesian and vice versa conversion in its the most challenging for someone as arithmetically challenged as me. While the formulas are not too hard to memorize its easily tricky and can get complicated at times. Especially with the presence of trigonometry

All in all i would say that my math 6 jourmey has been one full of learning. i am especially thankful for my teacher the one and only the man the myth the legend sir Jed de Leon!! As well as my fellow peers and classmates in math 6 who have helped me a lot throughout my journey. While its sad that this is the last blog of this account, the completion of this journey is definitely cause to celebrate! As well as the future that is to come in my mathematical misadventures!

[xyymath]

Symmetry: The beauty of Mathematics

Listen, symmetry isn’t just aesthetic. It’s the backbone of mathematics, the reason your equations behave, and honestly, the only thing keeping some functions from spiraling into utter chaos. And here’s the truth: symmetry isn’t just about looking pretty; it’s about mathematical harmony, balance, and the deeply satisfying moment when everything aligns perfectly, just as it should.

The Y-Axis Mirror Selfie (a.k.a Even Functions)

2. The Rotational Drama Queens (a.k.a Symmetry about the Origin; odd functions)

3. Group Theory Symmetry : the reason your Rubik's Cube works and why physicists don’t spiral into existential despair when studying quantum particles.

Cyclic groups, Dihedral groups and Lie groups

4. Polar Coordinates—aka the Art Students of Math

Polar graphs SCREAM symmetry. Roses? Circles? LEMNISCATES?

the bottom line : Symmetry is how we recognize order in chaos. And symmetry isn’t just confined to math. You see it in biology (DNA), art (da Vinci’s Vitruvian Man), and even in the way we naturally find faces with symmetry more attractive. It's everywhere, from the humble parabola to the complex eigenfunctions of quantum systems. So next time you plot a graph, take a second to appreciate the sheer elegance of its symmetry. Or lack of symmetry—because asymmetry can be just as hot.

[azurepolarbear]

⚫🔵🟣 #wip #codeart

The sine function is a trigonometric function whose output oscillates between one and negative one. Plotting the function in two-dimensional space results in a wave-like graph, whose appearance can change based on the length, amplitude, and frequency of the wave.

“rainbow waves v2.0” is an original generative code art algorithm; each run of the code produces a random visual output. Sine waves are created and stacked on top of each other. Each wave is made from a random number of points, with a random amplitude and frequency. Points are assigned their colors from a color selector that chooses colors from a random palette.

“rainbow waves v2.0” was made with TypeScript, p5.js, and the batpb/genart library.

The source code of this project is licensed under the GNU Affero General Public Version 3.0 License.

The visual output of this project is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) License.

Copyright (C) 2015-2024 brittni and the polar bear LLC. Some rights reserved.

[mathematics6weeklyblog]

The 3D Coordinate System

Moving out of the frigid landscapes of the polar coordinate system, we now have the mountains created by the 3D coordinate system, a topic not necessarily new to us with its applications in Physics 2, but still a stranger to me nonetheless. The amount of new concepts that were being thrown at us in this new topic was extremely overwhelming for me, especially since I was not gifted with natural depth perception to easily visualize 3D planes like my other friends in class. However, if Sir Dex masters the topic, and he admits to not having that level of depth perception either, then surely I can put in some time to answer practice problems related to the lesson.

Even in plotting points, I found it difficult to answer problems exploring all the concepts surrounding the 3D coordinate system, from distances to planes, to volumes of figures, to properties of spheres. However, most intimidating of them all are the graphs. For example, the graph of y = x^2 on the Cartesian plane would look entirely different on a 3D plane. This is not even incorporating equations where all three components (x, y, z) are present. Apparently, there was literally no tactics or tricks to graph the equations. It was simply visualization, which is extremely difficult for me to go through at the moment.

I am actually really worried about my results for the upcoming long tests. This was framed as the easiest lesson simply giving us bonus points for our grade, but now, I'm not so sure. Regardless, I'm not going to take my foot off the gas pedal now. I'm going to give it my all now to finalize my (hopefully) 1.00 final grade in math to end my year off.

Figure 3. Notes on the 3D Coordinate System (wow ayos ng notes ko dito haha)

[transiently-studying]

more about the plotting and then some about polar coordinates , this for you to mess with the graph yourself

wild how much you can learn in just 5 minutes

limacon

[biological-chemistry]

Part 1/2

This seminar, titled Social Media and Polarization, was delivered by Mathew Gentzkow of Stanford University about the about the causes of polarization (he also said he would offer solutions however they did not manifest). Gentzkowposes two key questions, 1) 'is the United States of America more polarized than ever before?', in short yes, dependent on the measurements used, and; 2) is social media to blame, he leads with the assumption that people generally believe this to be so. Building on this, he demonstrates past literature suggesting polarization to a social myth as recently as 2008. Personally, I never had the assertion the USA was more polarized than ever before. I feel Gentzkow assumed something of an international audience that might not translate. He might be referencing a media narrative that I am not privy to. I think considering one's audience's frame of reference is important while giving seminars, therefore in future presentations, I will consider which assumptions I present that are culturally based to foresee how they could be confusing to a foreign audience and try to explain them better.

Genntzkow leads the argument by examening the three areas whereedon't see increasing divisions. I found this to be a very honest way to lead the discussion, as it focuses on the limitations and scope of the evidence he will later produce. These areas are as follows: people's self-described ideologies, they dont self identify on the extremes of left or right; party identification, ergo the polticians similarly do not identify as far-left or far-right; and, views on some individiual issues, we dont see people taking more extreme policy views. These self-reporting results are likely to be biased, I am aware that the terms far-left and far-right have strong negative connotations therefore it makes sense the majority of people would llike to avoid being labeled thus. However, the fact people do not have more polerized policy views demonstrated the polarization in an objective sense does not come from public opinion. It is more likely a trend in mainstream media. I seems the Genztkow is activley avoiding the questioning cui bono.

Gentzkow, introduces us to the observations that show correlational evidence for increasing divisions in American voters, these are as follows: among issue views, party votes, and hostility/negative feelings about the opposite party. First, he uses a bar chart to illustrate the temperament of parties towards one another in 1960 and 2008. The results indicate people perceive the opposite party to be more selfish and of lesser intelligence in 2008 than they did in 1960. The comparison continues in showing that people in 2008 express more displeasure towards their children marrying someone of the opposite party, overtaking race which was a far more divisive issue in 1960. Although this is interesting, I don't find a comparison with just two dates strongly persuasive. Gentzkow's own work can be shown on a basic line graph, data from the 1970s to 2020s plot the divergence of affect towards ones own and the opposite parties. Over time people have become more critical of the opposite party and more loyal to their own.

[cyborgbuddy]

best shape? (2D, 3D, and 4D?)

anon i love you. i'm gonna get really nerdy with this one

favorite 2d shape has to be the cardioid

i like this one for a lot of reasons. i think it's really cool mathematically-speaking that one of the simplest functions you can plot in polar coordinates makes a heart-shaped graph. "simple love"

i think it also has a certain indescribable funny character to it, like if shapes were people this would be a george-from-seinfeld type shape.

favorite 3d shape is probably scutoids

crystalline structures and tessellations have always been really interesting to me. i think there's something special about a shape you can tile infinitely in all directions. the thing is, most of the standard shapes that can do this are pretty regular and simple. i like scutoids cause they can do this and they're not as simple

i'll use an analogy. cube and hexagonal crystalline structures are vanilla, focus-tested, conventionally attractive, Disney-produced Marvel Movie, andy warhol campbell chicken noodle shit. scutoids, they got that upper ear piercing, the big nose, the certain je ne sais quoi that someone especially attractive has, they're like a Wes Anderson movie if Wes Anderson was less mainstream, they're energetic and yet manage to keep this composition, it's like a gustav klimt or joan miro or van gogh

I also like the Boy's surface cause it's called the Boy's surface. projective planes are pretty cool in their own right of course

my favorite 4d shape doesn't really have a name but it's the spacetime manifold created outside the event horizon of a rotating black hole.

black holes that rotate create the jet streams you might see associated with black holes, and i think astronomically (besides closely related quasars) these are some of the coolest objects around. regular black holes, they're cool and all, they suck like a vacuum, they do the job and they do it well. rotating black holes, they do all that with an added tornado effect, they suck like how i imagine a 6'4" jk simmons-like guy who does mouth exercises sucks, and the result is spectacular

thanks