Consider the closed curve on the mobius strip that covers both sides of the fundamental polygon:   Is this curve viewed as a graph (with one vertex?) non planar since it must intersect itself in the plane? (help me make this more rigorous)  How do I more rigorously prove that certain graph in the mobius strip is non planar? Do I need to use some planarity criterion like Kuratowski's?
Is there an easy way to tell if a graph can be embedded on a Möbius strip (with no edges crossing)?  A specific version of this: if a simple graph with an odd number of vertices has all vertices of degree 4, can it be embedded on a Möbius strip?    Answer summary:   There are  for the Möbius strip that are   and drawn on . There are  for the Möbius strip, they are , and even better there are only .   I have not yet answered the second question, but Jim Belk's excellent explanation has given me good ideas.
In attempting to wrap my mind around the basic vocabulary, concepts, and methods of philosophy, I find myself wondering what the difference is between a philosopher and a theologian.   can have two meanings: 1. Theology is a rational study of the existence of God/gods and the nature of religious ideas. 2. Theology is simply a study of a particular religion (or all religions), really more the practice than the theory, but maybe a mixture.  It seems from an immediate reading that definition 1 would lead to the conclusion that a theologian is a philosopher (perhaps that theology is a branch or subset of philosophy), but definition 2 appears to be rather unphilosophical in nature.    So, how does philosophy answer this question?   Is theology considered a proper subset/branch of philosophy overall? Does it depend entirely on which of the above 2 definitions of theology fits a particular theologian as to whether he is a philosopher or not?   I think this answer is simple and fairly self-evident, but somewhere in my gut I feel like I'm missing something.  Is there a treatment of this question in the literature?    Edit: the answers so far seem to want to redefine theology or other terms - that is fine, but please be clear that you are redfining them, and provide, if possible, sources for why the term should be different than given (support from historical works, other sites, etc.). What is a tty, and how do I access a tty? How to format a duration in java? (e.g format H:MM:SS) I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity? How do I calculate the resistor value for a simple LED circuit? How to ‘guess’ if a noun is countable or uncountable? How to control RAM / swap usage Is there a way (application) to control RAM / swap usage policy at application level.  I have some applications that I use all the time and I want them to stay in RAM all the time and there are also application that I use quite seldom and they might be thrown to swap at demand. I know that default configured windows should give me best performance but I want to check whether I can get more :).  I use windows 7. How to do this manually, not installing any tools? I'm looking to book flights from) Philadelphia (PHL) to Detroit (DTW). Direct flights are quite expensive. I noticed, in expanding a search to surrounding airports, that I can book from Allentown (ABE) to Detroit for far less, and the connection is through Philadelphia.  Can I book this flight but board in Philadelphia? I'll be traveling with carry-on luggage only, so on the return, it's obviously no problem. But would I be allowed through security and on to the plane in PHL with out having flown from Allentown? Solder does not stick to tip of soldering iron? Basic question here, but it's driving me insane!  The solder does not seem to stick to the tip of my soldering iron. After the iron heats up, the tip appears black (sometimes on all sides and sometimes on one side), then either it doesn't melt the solder at all, or it I fiddle around with it for a while; it melts the solder, but the solder rolls into itself into a ball and doesn't stick to the tip.  I've seen this many times while soldering before and usually it's solved by cleaning the tip on a soldering sponge or cooling it down and scraping the tip off with a blade, but I have scraped this particular one many times and still it's the same problem: it heats up and gets black, then the solder doesn't stick to it.  It doesn't help that this soldering iron has a very fine tip and I have to use it to solder very tiny chips and wires under a microscope (I'm just starting to learn soldering under microscopes).  Any ideas on what I can do?    PS: Is scraping the tip a bad idea? I have been doing it with my other bigger soldering irons and it seemed to work (at least in most cases) but I am not sure whether it is a good solution or not. Do we have to tune the number of trees in a random forest? Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$? Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is:  $$ \nabla_A \operatorname{trace}( ABA^{T}C )  = CAB + C^T AB^T $$  There is a proof . However, I was not able to understand the second line of the proof. it says something like this:  $$ \nabla_A \operatorname{trace}( ABA^T C )= \nabla_\bullet \operatorname{trace}(f(\bullet)A^{T}C) + \nabla_\bullet \operatorname{trace}(f(A) \bullet^T C) $$  Where  $f(A) = AB$  I tried searching the proof or any hint using google, but I couldn't fine anything. Could anyone please help me with this second line. I cannot sleep. I am obsessed with this proof. Resizing Mac OS partition - macOS High Sierra How to select a clustering method? How to validate a cluster solution (to warrant the method choice)? One of the biggest issue with cluster analysis is that we may happen to have to derive different conclusion when base on different clustering methods used  (including different linkage methods in hierarchical clustering).  I would like to know your opinion on this - which method will you select, and how. One might say "the best method of clustering is which gives you the right answer"; but I may question in response that cluster analysis is supposed to be an unsupervised technique - so how do I know which method or linkage is the right answer?  In general: is a clustering alone robust enough to rely on? Or we need a second method and get a shared result to be based on both?  My question is not only about possible ways to validate / evaluate clustering performance, but is broader - on what basis do we select/prefer one clustering method/algorithm over another one. Also, is there common warnings that we should look around when we are selecting a method to cluster our data?  I know that it is very general question and very difficult to answer. I only would like to know if you have any comment or any advise or any suggestion for me to learn more about this. I would like to know how to put a time delay in a Python script. Put numbers in legend using pgf pie I would like to have the numbers that are inside the pie chart to be in the legend. How can I make that happen? Here is the MWE: \documentclass{article} \usepackage{pgf-pie} \usepackage{tikz} \usetikzlibrary{shadows}  \begin{document} \begin{tikzpicture} \pie[text=legend,style=drop shadow,radius=2]{51.3/Protestant, 23.9/Roman Catholic, 3.3/Other Christian, 1.7/Jewish, 0.7/Buddhist, 0.6/Muslim, 0.4/Hindu, 1.2/Other religions, 16.1/No religion } \end{tikzpicture} \end{document}  EDIT: I also have another pie chart, where the text is quite long. How can I set the width of the text so that it wraps around before the page ends? Here is what it looks like:  I forgot that PGF-PIE was not on CTAN. Here is a link to the google source: Ellipse formula from points Question: Best way to derive an ellipse formula when given bunch of points (say 60 points which when connected draws an ellipse)?  Background: Using R Momocs library function  which returns confidence ellipse coordinates. Would be nice to abstract those points into a formula.  You can see my ellipse points in the image below;    Any suggestions much appreciated, Thanks Importing 3D DXF file into QGIS with z information as attribute value? I have a dxf file of 3d contours. after import it does have only  layer,x,y attribute and doesn't have Z attribute.  How can I provide Z attribute to create a TIN file? Implementing switch cases I'd like to have command built around a switch-case environment in LaTeX, much like the example below:  \documentclass{article}  \usepackage{xstring}  \newcommand{\dothis}[1]{% \IfStrEqCase{#1}{{a}{so you typed a}     {b}{now this is b}     {c}{you want me to do c?}}     [nada] }  \begin{document}  \dothis{a}  \dothis{b}  \dothis{c}  \dothis{e}  \end{document}   My problem with it is that it requires the xstring package. Is there any other way to do this? Preferably without loading additional packages and avoiding disgraceful thickets of \if-\else-\fi statements? How to pass multiple operator properties via UI layout? Basically, I want to have a single operator which can do multiple things, based on some attributes values.  class MyPanel(bpy.types.Panel):     # (...)     layout = self.layout     row = layout.row()     row.operator("my.button", text="Button text")  class MY_BUTTON_OT_Button(bpy.types.Operator):     bl_idname = "my.button"     bl_description = "Button description"     bl_label = "Button"     foo = bpy.props.IntProperty()     bar = bpy.props.BoolProperty()      if bar:         # Do something with foo     else:         # Do something else :P   I know how to set one attribute, like so:      row.operator("my.button", text="Button text").foo=5   But what about multiple arguments? I tried several things that didn't worked, like (e.g.):   With setattr(): row.operator("my.button", text="Button text").setattr(foo=5,bar=True)   With a custom method setVal() in the operator class: row.operator("my.button", text="Button text").setVal(foo=5,bar=True)   Maybe I must use a single EnumProperty? I hope to avoid this, so my question is: Is it possible to pass multiple custom arguments to set attributes in an operator class? And if yes, how ?  Thanks for your help.