#### 45 cubics, series II (cylinder cut), 15cm

The complete series of 45 types of cubic surfaces with only finitely many singularities and lines. We provide links to put a complete set of several cubic surface models into your shopping cart.

The post Equations for lines and circles appeared first on mathematical sculptures by MO-Labs.

]]>*This is a post in our “equations and shapes” category, where we post articles explaining the relations between equations and the shapes they define. *

At school, lines and circles are some of the first geometrical objects you encounter. Later, you also learn that they may be described by equations defining points in a coordinate system. In this post, we review this relationship between lines and circles in 2d and equations briefly with a focus on implicit equations.

In subsequent posts, we will use this to go into 3d space, and explain equations for planes and spheres (i.e., surfaces of balls), and later even for much more involved shapes such as the famous Barth sextic world record surface.

From school, we know how to interpret equations such as in an -coordinate system. It defines a line with slope meeting the vertical axis in the point .

What you see above is not a static image, but an interactive app! Feel free to play with the parameters and to adapt the equation. Via the white point in the top right grid, you may change both and at the same time!

*Side note: It is beyond the scope of this post to explain how and why the parameters are linked to the line. Just very briefly: chaning the constant term moves the line up and down, the factor in front of the is the slope of the line. If you are interested in more background on this, just send me a message, and I will try to write a post on this in the future.*

Have you noticed that it is posssible to define a horizontal line in this way, e.g. ( does not appear), but that it is not possible to define a vertical line by a formula of the type in an -coordinate system? This is because the slope has to be “infinity” () for this special case. But we wish to avoid at this point. One way to do this is to note that a vertical line contains exactly those points which have a common -coordinate. So, we may describe all points on a vertical line with -coordinate by the equation: . In such an equation, does not appear.

In order to be able to treat even such cases without having to think about “infinity”, we may add another parameter in front of the , say , which may be zero or not. The complete equation now reads: . For , we get non-vertical lines as before, and for , we usually get vertical lines.

An even more special case occurs if and . For , this yields the equation which is true no matter which values of and we choose because these variables do not appear at all, so all points in the plane satisfy this equation! And for , say , we get an equation of the kind which is not true, no matter how we choose and .

All these cases may be experienced by playing with the parameters in the following app:

Similarly, for a polynomial of degree in one variable , say , we may consider the graph. It will be a so-called parabola. You may try to understand the effect of the parameters . In some cases, it is not difficult. E.g., changing will move the graph up and down. Can you describe the effect of parameter ? The exact effect of parameter is much more difficult to understand, I think.

*Side note: As above, we are not able to go into details about this here because this is standard school material und thus considered as basics for the purpose of this blog. But again, if you are interested in the basics behind this, just send me a message, and I will try to write a post on these basics in the future.*

In contrast to most of the curves above – but similar to vertical lines – circles may not be defined as a graph of a function of the form for some polynomial in one variable because for many values there are two different corresponding values of . But we may still describe any circle by a single polynomial equation thanks to the Pythagorean theorem:

**Theorem** (usually called “Pythagorean theorem” – to be discussed and proved in a subsequent post):

*A triangle with edges is rectangular with longest edge if and only if .
(In other words, the sum of the areas of the squares over the edges and is the same as the area of the square over .)*

How can we use this now in order to find an equation describing all points on a circle? Let us start with the simplest case where the center of the circle is the origin of the -coordinate system. Point in the picture has -coordinate and -coordinate – at least when is in the upper-right part of the coordinate system; otherwise, one or both coordinates will have an additional minus-sign. But in any case, we may use the Pythagorean theorem to conclude that . Now, as and , it follows for any point on the circle that , and is the radius of that circle.

In order to generalize this to the case of a circle of radius with a center not necessarily at the origin , note that for any point , we may consider the translated point . For the center of the circle, this will be the origin . For any point on the circle around , the translated point will be a point on a circle around the origin, of the same radius . Conversely, any point satisfying lies on a circle of radius around .

In subsequent posts in the “equations and shapes” category, we will use equations for lines, parabolas, and circles as described above for three dimensional objects.

This will go far beyond planes and spheres. One natural class of objects involving the equation of a circle are surfaces of revolution such as hyperboloids, vases, etc.

I hope in the future I will be able to extend my app *cwp* to allow me to realize some interesting problems you may solve directly within your browser.

Apart from the mathematics, another focus in this category of posts will be the fascinating people and historical developments behind some of the equations and shapes.

This post uses our own software *cwp* to visualize curves with parameters. It is based on the javascript library CindyJS. See Math-Sculpture.com/cwp for details. Other widgets have been created using pure javascript/CindyJS programming, without using our *cwp* app.

The post Equations for lines and circles appeared first on mathematical sculptures by MO-Labs.

]]>The post Equations and Shapes appeared first on mathematical sculptures by MO-Labs.

]]>*This is an introduction to our series of posts to be published in our “equations and shapes” category. In this series we post articles explaining the relations between equations and the shapes they define.
*

We will start by understanding very simple curves, and later advance to more interesting curves and surfaces. You will read about the famous cubic surfaces first studied in the 19th century, and you will even be able to learn about so-called world record surfaces such as Barth’s famous sextic.

Figure: A 3d-printed smoothed version of the famous Barth sextic world record surface, created by Oliver Labs. In this category of posts, you will read about how to create equations for surface such as this one.

This series of posts will focus on the following topics:

- equations,
- shapes,
- often both of these intertwined,
- visualizations / illustrations / 3d mathematical models / 3d math sculptures / … related to them.

And questions around this, such as:

- Where do/did they come from?
- Who found/created/studied them?
- How?
- Why?
- etc.

I am finally starting this series of posts around equations and shapes… a project I have been wanting to work on since my first advent calendar on beautiful algebraic surfaces in 2002, showing 24 images I created using the computer software “surf” (through my front-end “surfex”). That calendar showed beautiful surfaces, but the explanations were far too short.

This was long before our 2006 advent calendar (together with Hans-Christian Graf v. Bothmer) on geometrical animations – which was a large step forward because for each of the 24 days, we did not only create an image, but a mathematical animation.

In december 2007, I went one step forward again. For the imaginary 2008 project, I created several interactive visualizations within the computer software “surfer”, the first version of which was essentially an easier-to-use and nicer-to-look version of my software “surfex” from 2001. Besides entereing equations, the users were able to manipulate surfaces by changing certain parameters which I placed carefully into the equations in order to allow for interesting deformations of the original surfaces, such as the Barth world record sextic surface, to give an example.

I am starting this new series of posts in the month of December again, but this time, the project is intended to last much longer. On the other hand, I will, of course, not be able to publish one post per day, so the frequency is much lower than with first two projects mentioned above. Technically, this is yet another step forward now, the main advantage being that the interactivity now happens within the browser, so that no software has to be installed to experience the fascinating mathematics; it works even on smart phones and tablet computers thanks to the software CindyJS.

But this technical improvement is not the main aspect. More importantly, this format of posts finally gives the space and opportunity to explain enough details for each and every topic I want to speak about. Long ago, in 2002, I had intended to create a series of explanatory web pages, but this format of a blog is much better suited to the fact that one article will be written after the other; and CindyJS is a tool existing only since about 2016 and giving me the necessary technical background to create most of the illustrations.

Figure: A screenshot of my 2002 advent calendar of algebraic surfaces.

This series of posts uses our own software *cwp* to visualize curves with parameters. It is based on the javascript library CindyJS and it is currently actually not much more than an adaption of a CindyJS example, but may grow over time if I realize I need more features while creating this blog. See Math-Sculpture.com/cwp for details.

Other widgets are created using pure javascript/CindyJS programming, without using our *cwp* app, but sometimes using CindyJS plugins such as Cindy3d or CindyGL.

–> next post

The post Equations and Shapes appeared first on mathematical sculptures by MO-Labs.

]]>The post A Quintic with 15 Cusps appeared first on mathematical sculptures by MO-Labs.

]]>This quintic with 15 cusps is a so-called world record surface: No other quintic is currently known with more cusps. In fact, it is known that quintics cannot have more than 20 cusps, but it is not known if there are quintic surfaces with more than 15 cusps!

If you want to go for surfaces looking really singular then you will have to choose another production method, e.g. laser-in-glass. See MO-Labs.com for models of this type.

We currently provide only one size of this model: medium (14.9cm / 5.9in).

To purchase our 3d-printed modern version of this model use the “buy now” link provided below. You will be sent to our online shop at shapeways where you will be able to choose a size and color. Your object will then be 3d-printed just for you and sent to you in really short time where ever you are in the world.

This object was shown in the exhibition at Noordwikerhout, 2017, in the Netherlands.

The post A Quintic with 15 Cusps appeared first on mathematical sculptures by MO-Labs.

]]>The post 45 cubics, series II (cylinder cut), 15cm appeared first on mathematical sculptures by MO-Labs.

]]>*The photo above shows our “45 cubics series II (cylinder cut)” at Strasbourg university, France, in June 2015 (the 12cm (about 4.6in) version of our series). This was the first exposition of our new and more elegant series of cubic surfaces cut by cylinders instead of balls.
*

On this page, we present a 15cm (about 5.8in) version of this series of cubic surface sculptures.

Ccubic surfaces have been classified several times in the past. The first reasonable classifications were developed by Ludwig Schläfli in 1858 and 1863.

There is a modern version by Knörrer and Miller from 1987. Their method of classification yields almost the same result. The most interesting part consists of the classification of all irreducible cubic surfaces with only finitely many singularities and lines. They found 45 different types of these. Oliver Labs carefully created equations for each of those types such that all main features of each of the surface can be visualized in a single image or model, called “KM 01”, …, “KM 45”. Here is the article in PDF.

Our series “45 cubics series II (cylinder cut)” presents exactly this classification. As the name of the series suggests, each of the cubic surfaces has been cut by a cylinder in order to get a finite object. This gives the whole series a quite elegant look. This second series has been optimized in several other aspects as well. In particular, we designed all models carefully such that the shapes are as close as possible to the abstract mathematical surface. This means, e.g., that their walls are as thin as possible and the singularities are as close to a point as possible while still keeping the objects producible using 3d printing.

This complete series of all 45 cubics (cylinder cut) with only finitely many singularities and lines was first exhibited at the university of Strasbourg, France, starting from June 2015. Our older cubic surfaces series I (ball cut) was first exhibited at Lisbon from 2012 on.

Below, you may choose your preferred subset of the series of all 45 types of cubic surface models (series II (cylinder cut)). For your convenience, we provide links via which you may put a complete set of several cubic surface models into your shapeways shopping cart:

Put all 45 models from “45 cubics series II (cylinder cut)” into your cart (these are 45 different cubic surface models of a size of approximately 11.9cm); to check the current contents of your shopping cart, you may open your cart here:

If you are just looking for a few of these, here are 6 cubic surface sculptures, carefully selected by Oliver Labs. To put these 6 objects from “45 cubics series II (cylinder cut), 15cm” into your shapeways cart, just click the following link; to check the current contents of your shopping cart, you may open your cart here:

These are the 7 models included in this set:

- KM 01: a smooth cubic with 27 straight lines. This is NOT the Clebsch diagonal cubic, but a slightly more general cubic, so that that 27 lines have a more general intersection structure.
- KM 06: a cubic with 1 conical singularity (A1) and 21 straight lines. Comparing this model with number 01, one can see how the singularity forces 12 lines to coincide in pairs.
- KM 11: a cubic with 2 conical singularities (A1).
- KM 27: a cubic with 4 singularirities and 9 straight lines. We call this cubic “Cayley/Klein cubic” because it was Arthur Cayley who first studied this cubic in a publication, and it was Felix Klein who presented the first model of it and who based his deformation theory of cubic surfaces on this surface.
- KM 43: a cubic with 3 cuspidal singularities and 3 straight lines. We call this surface “Salmon’s cubic” because George Salmon was the first to realize that 3 is the maximum possible number of cusps on a cubic.
- KM 44: a cubic with 1 conical singularity (A1) and one higher biplanar singularity (A5) 2 straight lines. We call this surface “Schläfli’s cubic” because Schläfli was the first to realize that a cubic may contain an A5 and an A1 singularity, but no higher A-singularity, in particular no A6.

To put each of the 45 models from “45 cubics series II (cylinder cut)” invididually into your shapeways cart, you may use the following links:

- KM 01 (series II (cylinder cut)), 15cm
- KM 02 (series II (cylinder cut)), 15cm
- KM 03 (series II (cylinder cut)), 15cm
- KM 04 (series II (cylinder cut)), 15cm
- KM 05 (series II (cylinder cut)), 15cm
- KM 06 (series II (cylinder cut)), 15cm
- KM 07 (series II (cylinder cut)), 15cm
- KM 08 (series II (cylinder cut)), 15cm
- KM 09 (series II (cylinder cut)), 15cm
- KM 10 (series II (cylinder cut)), 15cm
- KM 11 (series II (cylinder cut)), 15cm
- KM 12 (series II (cylinder cut)), 15cm
- KM 13 (series II (cylinder cut)), 15cm
- KM 14 (series II (cylinder cut)), 15cm
- KM 15 (series II (cylinder cut)), 15cm
- KM 16 (series II (cylinder cut)), 15cm
- KM 17 (series II (cylinder cut)), 15cm
- KM 18 (series II (cylinder cut)), 15cm
- KM 19 (series II (cylinder cut)), 15cm
- KM 20 (series II (cylinder cut)), 15cm
- KM 21 (series II (cylinder cut)), 15cm
- KM 22 (series II (cylinder cut)), 15cm
- KM 23 (series II (cylinder cut)), 15cm
- KM 24 (series II (cylinder cut)), 15cm
- KM 25 (series II (cylinder cut)), 15cm
- KM 26 (series II (cylinder cut)), 15cm
- KM 27 (series II (cylinder cut)), 15cm
- KM 28 (series II (cylinder cut)), 15cm
- KM 29 (series II (cylinder cut)), 15cm
- KM 30 (series II (cylinder cut)), 15cm
- KM 31 (series II (cylinder cut)), 15cm
- KM 32 (series II (cylinder cut)), 15cm
- KM 33 (series II (cylinder cut)), 15cm
- KM 34 (series II (cylinder cut)), 15cm
- KM 35 (series II (cylinder cut)), 15cm
- KM 36 (series II (cylinder cut)), 15cm
- KM 37 (series II (cylinder cut)), 15cm
- KM 38 (series II (cylinder cut)), 15cm
- KM 39 (series II (cylinder cut)), 15cm
- KM 40 (series II (cylinder cut)), 15cm
- KM 41 (series II (cylinder cut)), 15cm
- KM 42 (series II (cylinder cut)), 15cm
- KM 43 (series II (cylinder cut)), 15cm
- KM 44 (series II (cylinder cut)), 15cm
- KM 45 (series II (cylinder cut)), 15cm

Below you will find other posts related to cubic surfaces:

The complete series of 45 types of cubic surfaces with only finitely many singularities and lines. We provide links to put a complete set of several cubic surface models into your shopping cart.

A math vase of degree 3 without bottom. It has been created by rotating a graph of a polynomial of degree 3 about an axis.

A math vase of degree 3. It has been creating by rotating a graph of a polynomial of degree 3 about an axis.

Cubic surfaces are a math model classic from the 19th century. We provide one of our favourite examples (cubic surface KM 42) in the form of a pendant.

The sculpture we present here is a 3D-printed modern object consisting of the 27 lines only, and a thin part of the surface as a border.

This visualizes a 1-parameter family of cubic functions or a 3d graph of a function in one variable in a 3d-coordinate system.

The post 45 cubics, series II (cylinder cut), 15cm appeared first on mathematical sculptures by MO-Labs.

]]>The post A gyroid, round cut appeared first on mathematical sculptures by MO-Labs.

]]>*The image above is not a photo, but it has been rendered by shapeways. See below our links to purchase it in our online-shop on shapeways.*

The gyroid is a modern classic. It is a so-called minimal surface. Our object is an approximation of it in terms of sine and cosine. One of its features is that it is triply periodic.

Often, people visualize a portion of this infinite object, but we think that a round cut is quite attractive as well. Especially when one is not so much interested in the infinite symmetries of the gyroid, but in its aesthetics.

Another version (with fewer “tunnels”) of this object (in white, size 162mm) is on display at

- the exhibition “3d print studio” in the Cuyperslab at the Cuypershuis, Roermond, Netherlands.

To purchase our 3d-printed modern version of this model use the “buy now” link provided below. You will be sent to our online shop at shapeways where you will be able to choose a size and color. Your object will then be 3d-printed just for you and sent to you in really short time where ever you are in the world.

The post A gyroid, round cut appeared first on mathematical sculptures by MO-Labs.

]]>The post Moebius Strip ring appeared first on mathematical sculptures by MO-Labs.

]]>*The Moebius Strip ring shown in the photos above are in the following materials: gold plated brass and premium silver. *

The Moebius Strip is a mathematical classic, and even for rings it has been used for many decades at least. Our version of the Moebius Strip ring, however, is special. Its shape is defined by a mathematical formula!

The Moebius Strip is one of the most famous shapes in mathematics, but also famous in real life. Its extraordinary “one-sided” shape stands for “infinity”, “everlasting friendship”, etc. The Moebius Strip may be realized by taking a long rectangular strip of paper, turn one of the two shorter edges by 180 degrees, and then glue the two shorter edges. In this way, when starting to “walk” at the glueing position on one side of the object, and continuing to walk on that side, one eventually ends up at the same point of the Moebius strip, but on the other side! Without having crossed the boundary of the shape… Thus, the Moebius strip actually only has a single side!!!

Our Moebius Strip ring is available in many ring sizes. If you cannot find the size you are looking for here, feel free to contact us, e.g. via our website Math-Sculpture.com.

Note that the colors provided by the Shapeways renderings are far from perfect. For some of the materials, we are able to provide real photos so that you may see how they actually look.

To purchase our mathematical formula version of the Moebius Strip ring use the “buy now” link provided below. You will be sent to our online shop at shapeways where you will be able to choose a material and ring size. Your object will then be produced just for you and sent to you in really short time where ever you are in the world.

The post Moebius Strip ring appeared first on mathematical sculptures by MO-Labs.

]]>The post Moebius Strip pendant appeared first on mathematical sculptures by MO-Labs.

]]>*The Moebius Strip pendant shown above is available in various materials: from colored plastics (such as the photo), over brass and bronze to silver, gold, and more.
*

The Moebius Strip is a mathematical classic, and even for pendants it has been used for many decades at least. Our version of the Moebius Strip pendant, however, is special. Its shape is defined by a mathematical formula!

The Moebius Strip is one of the most famous shapes in mathematics, but also famous in real life. Its extraordinary “one-sided” shape stands for “infinity”, “everlasting friendship”, etc. The Moebius Strip may be realized by taking a long rectangular strip of paper, turn one of the two shorter edges by 180 degrees, and then glue the two shorter edges. In this way, when starting to “walk” at the glueing position on one side of the object, and continuing to walk on that side, one eventually ends up at the same point of the Moebius strip, but on the other side! Without having crossed the boundary of the shape… Thus, the Moebius strip actually only has a single side!!!

Note that the colors provided by the Shapeways renderings are far from perfect. For some of the materials, we are able to provide real photos so that you may see how they actually look.

To purchase our mathematical formula version of the Moebius Strip pendant use the “buy now” link provided below. You will be sent to our online shop at shapeways where you will be able to choose a material and/or color. Your object will then be produced just for you and sent to you in really short time where ever you are in the world.

Here are some more posts related to Möbius:

The Moebius Strip is a mathematical classic, and even for rings it has been used for many decades at least. Our version of the Moebius Strip ring, however, is special. Its shape is defined by a mathematical formula!

The Moebius Strip is a mathematical classic, and even for pendants it has been used for many decades at least. Our version of the Moebius Strip pendant, however, is special. Its shape is defined by a mathematical formula!

The Moebius Strip is a mathematical classic, and even for rings it has been used for many decades at least. Our version of the Moebius Strip pendant, however, is special. Its shape is defined by a mathematical formula!

The Moebius strip is a simple, but fascinating math object, with just one side and one connected boundary curve. Werner Boy's surface contains such strips!

The post Moebius Strip pendant appeared first on mathematical sculptures by MO-Labs.

]]>The post Moebius Strip pendant appeared first on mathematical sculptures by MO-Labs.

]]>*The Moebius Strip pendant shown is not an ordinary Moebius Strip – which is a thin rectangular strip, turned by 180 degrees (i.e. a half turn) on one of the short edges, and then the two short edges glued -, but the the thin edge has been turned by 3*180 degrees (i.e. three half turns).
*

The Moebius Strip is a mathematical classic, and even for pendants such as this one, it has been used for many decades at least. Our version of the Moebius Strip pendant (3 half turns), however, is special. Its shape is defined by a mathematical formula!

The Moebius Strip is one of the most famous shapes in mathematics, but also famous in real life. Its extraordinary “one-sided” shape stands for “infinity”, “everlasting friendship”, etc. As mentioned above, the Moebius Strip may be realized by taking a long rectangular strip of paper, turn one of the two shorter edges by 180 degrees, and then glue the two shorter edges. In this way, when starting to “walk” at the glueing position on one side of the object, and continuing to walk on that side, one eventually ends up at the same point of the Moebius strip, but on the other side! Without having crossed the boundary of the shape… Thus, the Moebius strip actually only has a single side!!!

The pendant presented here is also a one-sided shape although it does not represent an ordinary Moebius Strip, but our Moebius Strip pendant has the short edge turned by three half turns before the “glueing”.

Our Moebius Strip pendant is available in many ring sizes. If you cannot find the size you are looking for here, feel free to contact us, e.g. via our website Math-Sculpture.com.

Note that the colors provided by the Shapeways renderings are far from perfect. For some of the materials, we are able to provide real photos so that you may see how they actually look.

To purchase our mathematical formula version of the Moebius Strip pendant use the “buy now” link provided below. You will be sent to our online shop at shapeways where you will be able to choose a material and ring size. Your object will then be produced just for you and sent to you in really short time where ever you are in the world.

The post Moebius Strip pendant appeared first on mathematical sculptures by MO-Labs.

]]>The post A math vase of degree 3 without bottom appeared first on mathematical sculptures by MO-Labs.

]]>This math vase of degree 3 is a classic from school teaching.

Indeed, it is just the graph of the function y=(x+9)*(x-68)*(x-68)*0.0003+10, rotated about the x-axis. Such math models are common examples for computing the area or the volume of a surface of revolution given by rotating a graph about an axis.

To purchase our 3d-printed modern version of this model use the “buy now” link provided below. You will be sent to our online shop at shapeways where you will be able to choose a size and color. Your object will then be 3d-printed just for you and sent to you in really short time where ever you are in the world.

You may purchase this math vase of degree 3 via the “buy now” button below:

We also created a version of this vase with bottom which you may actually use to measure the amout of water that fits into it (although the “strong & flexible” material is not really watertight). Some more school math sculptures are listed below:

A math vase of degree 3 without bottom. It has been created by rotating a graph of a polynomial of degree 3 about an axis.

A math vase of degree 3. It has been creating by rotating a graph of a polynomial of degree 3 about an axis.

This visualizes a 1-parameter family of cubic functions or a 3d graph of a function in one variable in a 3d-coordinate system.

The post A math vase of degree 3 without bottom appeared first on mathematical sculptures by MO-Labs.

]]>The post A math vase of degree 3 appeared first on mathematical sculptures by MO-Labs.

]]>This math vase of degree 3 is a classic from school teachingT

( Attention: The “white, strong & flexible” material is not 100% water proof. However, a brief experiment in school for measuring the volume, works fine. )

Indeed, it is just the graph of the function y=(x+9)*(x-68)*(x-68)*0.0003+10, rotated about the x-axis. Such math models are common examples for computing the area or the volume of a surface of revolution given by rotating a graph about an axis.

You may purchase this math vase of degree 3 via the “buy now” button below.

Once again: Attention! The “white, strong & flexible” material is not 100% water proof. However, a brief experiment in school for measuring the volume, works fine.

We also created a version of this vase without bottom which has some advantages for teaching purposes. Some more school math sculptures are listed below:

A math vase of degree 3 without bottom. It has been created by rotating a graph of a polynomial of degree 3 about an axis.

A math vase of degree 3. It has been creating by rotating a graph of a polynomial of degree 3 about an axis.

This visualizes a 1-parameter family of cubic functions or a 3d graph of a function in one variable in a 3d-coordinate system.

The post A math vase of degree 3 appeared first on mathematical sculptures by MO-Labs.

]]>