Cube Definition and 610 Threads

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.
The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
The cube is dual to the octahedron. It has cubical or octahedral symmetry.
The cube is the only convex polyhedron whose faces are all squares.

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  1. M

    Are these values of the following correct? (Complex cube root of unity)

    Given that x is a complex cube root of unity, we have ## x^{3}=1 ## but ## x\neq 1 ##, where the cube roots of unity are ## 1, \omega, \omega^2 ## and ## \omega, \omega^2 ## are the imaginary roots such that ## \omega^3=1, 1+\omega+\omega^2=0 ##. Now we will consider three cases of the...
  2. T

    Resonance of a cube floating on water under external force

    My attempt is approaching this problem like the mass spring model. Considering the buoyancy force as spring force. By doing so, we can have the typical mass-spring equation 𝑚 (𝑑^2 𝑥)/(𝑑𝑡^2 )+Fbuoyancy = 𝐹_𝑒𝑥𝑡 Then I can assuming the displacement a will be the sinusoidal function...
  3. S

    B What would need to be possible to make a cube of circles?

    in what geometry would a cube of circles be possible
  4. Ranger Mike

    Question about ice cube tray freezing first in the front

    why is it, the ice cube tray freezes first, at the front of the tray closest to the door. This should be the warmest place of the refrigerator. The back does not have enough time to send the colder air out to atmosphere. I think i know the answer but lets us explore this..
  5. paulimerci

    Find the magnitude of the electric force from 3 charges at vertices of a cube

    There are three charges with +1 μC and −1 μC, are placed at the opposite corners of a cube with edges of length 1 cm, and the distance from P to B is 1cm 2. I labeled them as A, P, and B, which is shown in the diagram below. Since we need to find the magnitude of the charge at point P and the...
  6. J

    How does the presence of a cube resistor affect current flow in a circuit?

    I would think that it is 2 A (6/3 because there are three wires the current can go to), but this does not seem to be correct.
  7. S

    Maximum angle reached by a cube placed inside a spinning cylinder

    I am trying to solve a problem where we have to find the maximum angle before a cube starts sliding when said cube is placed inside a spinning hollow cylinder (the cylinder is placed horizontally). The radius of the cylinder is 0.4 m, the coefficient of static friction between the cube and the...
  8. WMDhamnekar

    MHB Evaluating $\iint\limits_{\sum} f \cdot d\sigma $ on Cube S

    Without using the Divergence Theorem Evaluate the surface integral $\iint\limits_{\sum} f \cdot d\sigma $ of $f(x,y,z) = xi+ yj + zk , \sum: $ boundary of the solid cube S= $\{(x,y,z) = 0\leq x,y,z \leq 1)\}$ My attempt: Here we have to use the following definition of surface integral. Note...
  9. jack action

    What lazy engineers achieve: A Rubiks cube that solves itself

    How much work does a lazy engineer is willing to put to NOT do the job? That much: source: fb.watch/e1LXE0nqqP/
  10. M

    No integer whose digits add up to ## 15 ## can be a square or a cube

    Proof: Let ## a ## be any integer. Then ## a\equiv 0, 1, 2, 3, 4, 5, 6, 7 ##, or ## 8\pmod {9} ##. This means ## a^{2}\equiv 0, 1, 4, 9, 7, 7, 0, 4 ##, or ## 1\pmod {9} ## and ## a^{3}\equiv 0, 1, 8, 0, 1, 8, 0, 1 ##, or ## 8\pmod {9} ##. Thus ## a^{2}\equiv 0, 1, 4 ##, or ## 7\pmod {9} ## and...
  11. M

    Given an integer, find the probability its cube ends in 111

    The last three digits of ##x^3## must be solely dependent on the last 3 digits of ##x##. So let ##x=a+10b+100c## for integers ##a,b,c##. Then ##x^3 = a^3 + 30 a^2 b + 300 a b^2 + 300 a^2 c +O(1000)## where of course ##O(1000)## don't affect the last 3 digits. Evidently ##a^3## is the only...
  12. G

    Flux of the electric field that crosses the faces of a cube

    a) $$\phi_T=\phi_F-\phi_I=10^4\cdot 4\cdot 4-10^4\cdot 4\cdot 4=0\, \textrm{Nm}^2/\textrm{C}$$ b) $$\phi_F=\underbrace{300\cdot 4}_{\vec{E}}\cdot \underbrace{4\cdot 4}_{\textrm{area}}=19200\, \textrm{Nm}^2/\textrm{C}$$ $$\phi_0 = 300\cdot 0\cdot 4\cdot 4=0\, \textrm{Nm}^2/\textrm{C}$$ Then...
  13. MatinSAR

    Electrical flux passing through the cube

    Picture for better understanding. My answer : I want to know how to solve this problem without using cylindrical. I mean how can we solve this using cube and its sides. Thanks.
  14. P

    Comparing Cube A & B's Temperature Change

    I know in this case cube A would heat up faster, but would cube B eventually reach the same temperature?
  15. M

    B Why Is a Cubic Polynomial Called 'Third Degree'?

    Why is a third degree polynomial called a cubic polynomial? I just don’t see the connection between 3 and a cube.
  16. antythingyani

    Averaging the cube of semimajor axis to position ratio wrt to time

    Summary:: Averaging (a power of) semimajor axis to position ratio wrt to time - celestial mechanics I evaluated it this far, but i don't know how to change the dt to d theta ... the final solution is supposedly (1-e^2)^-(3/2) . Any help will be appreciated. [Image re-inserted with correct...
  17. D

    Momentum of Cube: Magnitude and In-Between Speed

    A) and b) should be useful for solving the initial question. If the truck is at rest initially, the magnitude of the momentum of the ball becomes ##|mv'|=|MV' - mv|##, but this may or may not be less than the magnitude ##mv##, depending on how large ##V'## is. ##V' = \frac{m(v+v')}{M}## in this...
  18. V

    Electric Flux for a cube problem

    I have tried to understand the solution given in the book which is as pasted below. The solution uses Gauss's Law but makes no mention of which Gaussian surface is used. The diagram that I have used to understand this problem is also given at the end. From the diagram, faces OADG, OABE and OEFG...
  19. P

    Misc. Display built with a rotating cube design

    Hello everyone, I want to build a rotating display cube (will be mounted on a wall, those cubes that has shops logos ect..), my problem is that I don't know where to start and how to attack the rotating mechanism. I have a 1400Rpm moteur (and it's a 1m per 1m cube that weights around 120kg-150kg...
  20. rudransh verma

    Flux of a non-uniform E field through the face of a cube

    Phi = int of (3xi^ + 4j^).vector dA = 3 int of(xdA) Now we put x= 3 and we get at last 36 N m^2/C. I am getting confused why E is a fuction of x. How can that be? How can we represent the E and position x on the same coordinate system. Is it right ? Because we know distance is inversely...
  21. P

    Magnetic field - General current in a cube

    I could solve a similar (rather, a specific case of the above) where the current entered through a corner and left from the corner opposite to it along the body diagonal of the cube. For this specific case, I was able to easily exploit symmetry to deduce the answer (0). However, I cannot think...
  22. V

    Hydrostatic Equilibrium Problem Involving a Cube and two Liquids

  23. Eagle9

    So, can we see a cube instead of sphere?

    <mentor - Epistemology (how do we know or perceive) is not subject PF supports in the science forums - moved to General Discussion> Since my previous topic The arrangement of our visual system and the objective truth was closed I will open a new one, less philosophical. So, just imagine that...
  24. DaveC426913

    B Angle of the diagonal of a cube

    I've looked for this in a hundred places but I keep finding the unit length of the diagonal (root 3). I want to find the angle. i.e. the angle between the cube's diagonal and its three edges at a vertex (all three angles will be identical, of course). I guess I hoping to figure it out without...
  25. Eclair_de_XII

    Proving continuity of inverse cube function

    The proof is given in two steps 1. Prove the lemma. 2. Use lemma to prove result. %%1-Lemma%% Assume ##a\neq0##. Define ##g:(-(|a|+1),|a|+1)\longrightarrow \mathbb{R}## by ##g(x)=\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}##. Then ##g## is bounded from below by some positive number ##m##...
  26. Averagesupernova

    Is My 1980s Rubik's Cube a Knockoff?

    As the title implies, I have a cube that was given to me new around 1980-1981. All the info I have found online indicates my cube is likely not an original. I would have thought at that time there wouldn't have been knock offs, but it's entirely possible I am wrong. - The colors on my cube do...
  27. M

    MHB Symmetry Operations of a Cube: Geometric Descriptions and Matrix Representations

    Hey! :giggle: Let $\displaystyle{W:=\left \{\begin{pmatrix}x\\ y\\ z\end{pmatrix}\in \mathbb{R}^3\mid x,y,z\in \{-1,1\}\right \}}$. Draw the set $W$ in a coordinate system. Let $v=\neq w$ and $v,w\in W$. If they differ only at one coordinate connect these points by a line. With this...
  28. TheMercury79

    Is the Cube of matrix associative?

    But I actually don't get the same matrix. What I get is the transpose of the other when I change the order i.e when I do [A]^2[A] I get the transpose of [A][A]^2 and vice versa What I'm trying to do is find the cube of the expectation value of x in the harmonic oscillator in matrix form. We're...
  29. AN630078

    Iterative root finding for the cube root of 17

    Firstly, the cube root of 17 is 2.571281591 which is 2.57 to 3.s.f. Initially, I thought about just approaching this problem using the Newton-Raphson Method when x0=2. In which case; x^3=17 x^3-17=0 Using the Newton-Raphson iterative formula xn+1=xr-f(xn)/f’(xn) f(x)=x^3-17 f’(x)=3x^2...
  30. S

    Pressure at the bottom of a cube immersed in two liquids

    I am not sure about value of depth I need to use. What I did: Hydrostatic pressure at bottom of cube = hydrostatic pressure by oil + hydrostatic pressure by water = ρoil . g . d + ρwater . g . h Is it correct I use value d = 0.2 m for depth of oil and value of h = 0.02 m for depth of water to...
  31. anemone

    MHB Sum of Numbers on Cube Faces to Equal 2004

    Positive integers are written on all the faces of a cube, one on each. At each corner (vertex) of the cube, the product of the numbers on the faces that meet at the corner is written. The sum of the numbers written at all the corners is 2004. If $T$ denotes the sum of the numbers on all the...
  32. jaychay

    MHB Find Point D on Plane for 4 Unit Cube Pyramid

    Find point d on the line of r(t)=(0,0,0)+(−1,1,1)t which make the triangular pyramid abcd has the volume of 4 unit cube when a(0,0,0),b(1,0,1),c(0,1,0) are the points on the plane of −x+z=0.
  33. bob012345

    B Cuboids in a Cube: {##1,n##} Dim. n

    Extending the previous problem of Cubes Within Cubes, where we asked how many cubes can be drawn within a cube, now we ask how many cuboids of size {##i ,j ,k = 1,n##} can be drawn in a cube of dimension ##n## where ##n## is an integer.
  34. Helios

    B Can a Square be Dissected into a Cube with Fewer Pieces?

    Dear Recreational Geometry People, I recovered a thing I did very long ago from a drawing of mine that I fortunately just found again. With some effort I was able reconstruct what I did and redraw it. It is a geometric dissection. The task is to slice up a square and use those pieces to make a...
  35. archaic

    Cube with charges at each vertex

    For convenience, I take the center of the upper face. The charges at the top cancel each other's effects, and those at the bottom cancel each other's horizontal effects, so I get$$E=4k\frac{q}{r^2}\sin\theta$$I have found that ##\theta=\arcsin\left(\frac{s}{r}\right)=\arcsin{\sqrt\frac{2}{3}}##...
  36. T

    Radiation from a cube at 189 °C

    I have tried to use the above equation. P2/64=(189/102)^4 P2=754.4431 But I don't know why the answer is wrong. Please help!
  37. S

    Flux due to a charge located at the corner of a cube

    The correct answer is B, but I am not sure why. I have a few confusions regarding this problem. First of all, I had thought that we cannot use Gauss' Law to determine the flux through a SIDE of a cube since Gauss' Law only works for SURFACES. How can we determine how an electric field pierces a...
  38. jisbon

    Solving for Maximum Speed of Cube: Kinematics & Upthrust

    So, I recognise that: $$ma=pg\left( L^{2}\right) \left( L-y\right) -\dfrac {1}{2}pgL^{3} $$ whereby $$pg\left( L^{2}\right) \left( L-y\right)$$ is the upthrust while the other is mg. So, to find the largest speed of cube, I will assume that acceleration is zero since the acceleration slowly...
  39. hnnhcmmngs

    Calculating Collision Frequency of Electrons in Copper Cube

    Homework Statement: Verify the claim of Section 7.2 that the electrons of a metal collide with the surface at a rate of about 10^30 per second per square centimeter. Do this by estimating the collision frequency of electrons in a 1.00-cm cube of copper metal with one face of the cube surface...
  40. M

    Find the fractional increase in inertial mass when an ice cube melts

    Summary: Apparently an ice cube gains mass when it melts So I'm asked to "Find the fractional increase in inertial mass when an ice cube melts ". All I've got off the top of my head right now is that a cube has energy = mc^2, and then when the cube melts, energy Q = (Heat of fusion)(m) is...
  41. X

    Cube root long division method

    On the right paragraph it says "The trial divisor 1200 goes into the dividend 13952, 8 times" Clearly 1200 goes into 13952, 11 times. I don't understand why 8 is (arbitrarily?) chosen. Please help. Thanks.
  42. M

    Net charge contained by a cube in a region with a non-uniform E field

    I'm having a little trouble understanding how to go about solving this problem. I was in class Tuesday and the hint I got from the T.A. running my discussion section was that : "because the electric field is only non-uniform along the x axis, the electric field will both enter(negative flux) and...
  43. Monoxdifly

    MHB [ASK] Find the volume of Pyramid in a Cube

    Given a cube ABCD.EFGH whose side length is 4 cm. If the point I, K, and J is dividing EF, FG, and BG to two equal lengths respectively, determine the volume of pyramid D.IJK! I think I can work out the pyramid's base area by deriving for the formula of equilateral triangle area. What I can't...
  44. andrea1313

    Black cube, maximal and minimal value of equilibrium temperature T

    So i had this problem and I want a rigourous solution. The answer should be : Tmin=(I/sigma)^(1/4) and Tmax=(sqrt(3I)/sigma)^1/4
  45. P

    I If I had a cubic metre of solid osmium, a perfect cube....

    If i shon a red laser across the surface of the osmium cube 5mm above the solid perfect 1000mm cube, by how many degrees would it be deflected?
  46. Moara

    Electric field of a charged cube

    Tried to use gauss law but there isn't any usefull symmetry that I have seen. Also tried to integrate the field due to small charges over the whole cube, didn't work too since the integral were too much complicated.
  47. D

    How to find the center of mass of a cube?

    Homework Statement Find the center of mass of a homogeneous solid cube with side ##L## analytically. Homework Equations None. The Attempt at a Solution I don't understand how to find the center of mass on three dimensions. I know that since it is homogeneous, if I center the cube on the...
  48. R

    Constructing a cube with a Norm

    Homework Statement Let X = ##\mathbb{R^m}## and ||.|| be a Norm on X. The dual norm is defined as ##||y||_*:=sup({\langle\,x,y\rangle :||x|| \leq 1})## a) Show that ##||.||_*## is also a norm b) Construct two norms ##||.||^O## and ##||.||^C## so that: {##x:||x||^O=1##} is a regular octahedron...
  49. Arman777

    The mean value of the cube, Force Field Laplace equation

    Homework Statement I have a value of $$ U=U_0+x (∂U/∂x)+y(∂U/∂y)+z (∂U/∂z)+1/2x^2(∂^2U/∂x^2)+1/2y^(2∂^2U/∂y^2)+...$$ We need to find the mean value of the U. So the answer is $$\overline{\rm U}\approx U_0+a^2/24(∇^2U)$$Homework Equations $$\overline{\rm U}=1/a^3 \int \int\int Udxdydz$$ The...
  50. MedEx

    Calculate Net Charge Contained by the Cube

    Homework Statement The figure shows a closed Gaussian surface in the shape of a cube of edge length 3.20 m. It lies in a region where the electric field is given by E=(1.65x+2.86)i + 4.39j + 5.47k N/C, with x in meters. What is the net charge contained by the cube? Homework Equations net...
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