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MATEMATICAS: DRAWING THE HIPERCUBE (YOU TUBE)
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Reply  Message 1 of 25 on the subject 
From: BARILOCHENSE6999  (Original message) Sent: 17/03/2013 03:07
rawing the Hypercube # 1 - YouTube
www.youtube.com/watch?v=ccws454YiVM
14/08/2008 - Subido por ProfessorElvisZap
Professor Zap shows how to draw a hyper cube in real time.
 
  • Hypercube Animation - YouTube

    www.youtube.com/watch?v=NE9ZFDJ4Phk
    06/12/2009 - Subido por brethilaki
    Video Responses · Thumbnail How to Draw a Hypercube (Tesseract) with 8 Squares by BRyanS72 2,662 ...
     
  • Discover The 4D. The Impossible Hypercube. - Video

    www.metacafe.com/.../discover_the_4d_the_impossibl...
    03/06/2007
    I am drawing there the HYPERCUBE; a cube that is impossible to imagine in a 3D world. Th. Watch Video ...


  • First  Previous  2 to 10 of 25  Next   Last 
    Reply  Message 2 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:31
     
     
    HIPERCUBO EN 4 DIMENSIONES 16 PUNTOS
     
    HIPERCUBO EN 5 DIMENSIONES 32 PUNTOS
     
    6 DIMENSIONES 64 PUNTOS

    Reply  Message 3 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:33
    5 DINENSIONES

    Reply  Message 4 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:35
    HIPERCUBO EN FORMA FISICA

    Reply  Message 5 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:41

    Hypercube Representations-2

    Magic Stars Home Misc. Numbers Home Site Home Page Magic Squares Home Magic Cube Home

    The previous page considered ways to present magic squares and cubes (2-D and 3-D objects).

    This page considers magic tesseracts (4-dimensional objects).

    Historical representations             Hendricks modern form             The first tesseract

                   rule-w.gif (2726 bytes)

    Historical representations

    • Magic squares are 2-dimensional objects consisting of 4 edges that meet 4 corners at right angles. They are easy to illustrate on a 2-dimensional surface.
    • Magic cubes are 3-dimensional objects consisting of  12 edges meeting 8 corners at right angles . They are easy to see in 3-dimensional space, but can be illustrated on a surface (2-D)  only by introducing distortion.
    • A magic tesseract is a 4-dimensional object consisting of  32 edges meeting 16 corners at right angles. A model of a tesseract could be built in 3-dimensional space, but only by introducing distortion (the corners would not be right angles). A drawing of a tesseract obviously requires still more distortion!
    In trying to visualize a tesseract, 2 types of drawings have been used for many years. This one shows a small cube 'suspended' inside a large cube and 'supported' by six distorted cubes.

    This is the best illustration to show how each tesseract is 'bounded' by 8 cubes (6 of them are distorted).


     
    This drawing illustrates the second visualization (the figure to the right).

    The drawing as a whole attempts to show how a hypercube may be 'dragged' through the dimensions to produce a hypercube illustration for each. i.e here we go from 0-dimension to 1, 2, 3, and finally arrive at the forth dimension.

    The numbers help to identify the corners. As an 'extra', I have arranged the numbers so that each square (and rhomboid) is perimeter magic. Remember that in these two cases, the rhomboids are actually distorted squares!

    In both examples, the drawing is obviously not suitable to display the numbers in the magic tesseract. Even for order-3, the lowest, there are 81 numbers to display.

    rule-w.gif (2726 bytes)

    Hendricks modern form

    In 1962, John R. Hendricks published a new method of drawing the magic tesseract [1][2].  He did it in grand style by describing a 6-dimension magic hypercube of order-3. This hypercube used the numbers from 1 to 729 (729 = 36) and required 9 order-3 dimension 4 (tesseract) figures to display (a 6-D figure was just too complicated to comprehend). Each tesseract sums on its own in 4 ways. The fifth direction is found by jumping from tesseract to tesseract horizontally, and the sixth direction by jumping vertically. The resulting normal 6-dimensional magic hypercube sums to 1095 in at least 1490 ways (6m5 + 32).
    He also showed a 5-dimension order-3 magic hypercube. It required three order 3 tesseract diagrams. See this hypercube here.

     

    He introduced the subject with figures 1, 2, and 3; showing an order 3 magic square, magic cube, and magic tesseract. Notice that none of the three magic figures are normalized. He came up with that idea (for cubes and tesseracts) at a later date, when he realized that a system was required for listing solutions in order. This tesseract is an aspect of index # 5.
    The magic square has traditionally been illustrated with each number occupying a 2-dimensional 'cell', not as intersections of a 'grid' as suggested in Figure 1 (above). The magic cube was also shown that way at the start, but now is normally shown with the numbers placed at grid intersections.

    This tesseract diagram is now the one in universal use for displaying small orders of  4-dimension magic hypercubes.
    For the larger orders, a simple text listing is still the most practical.

    This is one way of listing the above tesseract

    09   76   38      64   35   24      50   12   61
    46   17   60      05   75   43      72   31   20
    68   30   25      54   13   56      01   80   42
    
    74   45   04      33   19   71      16   59   48
    15   55   53      79   41   03      29   27   67
    34   23   66      11   63   49      78   37   08
    
    40   02   81      26   69   27      57   52   14
    62   51   10      39   07   77      22   65   36
    21   70   32      58   47   18      44   06   73

    The purpose of this  page is to show methods of illustrating a magic tesseract. Other pages on this site will discuss features and characteristics of the magic tesseract, and the relationships between this 4-d magic figure and it's 2-D and 3-D cousins.)

    [1]John R. Hendricks, The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, vol. 5, no. 2, 1962, pp 171-189
    [2]John R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1998,  0-9684700-0-9  the preface contains some history of this discovery

    rule-w.gif (2726 bytes)

    The first magic tesseract?

    In 1905, Dr. C. Planck published a paper [1] [2] for private circulation. Called The Theory of Paths Nasik, It dealt with perfect magic squares and cubes.

    Dr. A. H. Frost defined the term Nasik in 1878 [3] as requiring that all paths sum correctly. He stated that the smallest order Nasik magic square is order-4 ( we call it pandiagonal), and the smallest order Nasik magic cube is order-8.

    In explaining his theory, Dr. Planck constructed the order-9 pandiagonal magic square shown in 2. He then took the nine order-3 sub-squares and arranged them as shown in illustration 1. He called this a 'crude-magic octahedron.

      1
     
    2
    as an order-9 pandiagonal magic square
    3
    another aspect of the same magic tesseract
     hh-1 (to the right) is the modern method of displaying the 4-dimension magic hypercube.
    All lines parallel to the edges, and also the 8 quadragonals, sum correctly to the constant 123.

    I quote from his paper

    ...we shall obtain the three cubes of order 3 shown in figure 11, which form a perfect crude-magic octahedroid of order 3. These three cubes are sections of the four-fold made by three parallel equidistant spaces, laid out in perspective in one space of three dimensions.

     Some points

    • The 3x3 squares in figure 11 are not magic, nor are the 3x3x3 cubes that may be formed from them. This is because the diagonals and triagonals do not sum correctly. This is not a requirement of a magic tesseract. The eight Quadragonals of the tesseract (such as 47+41+35) is a requirement and does sum correctly.

    • Figure 11a is shown in green in the modern drawing with the other two cubes parallel to it. The three cubes of fig. 12 are parallel to the front of the drawing.

    • This tesseract is 1 of 384 aspects of the 58 basic magic tesseracts of order 3. It is a non-normalized aspect of index number 57 (of the 58).

    • Usually, to make the drawing simpler, only the outline lines are shown (see previous section).

    • hh-2 is another aspect of the same 'octahedroid'. Do you see where Planck,s cubes fit in this drawing?

    Perfect?

    Planck's paper expanded on Frost original definition of Nasik, applying it to hypercubes where all lines sum correctly i.e. perfect hypercubes.
    I must emphasize, though, that this tesseract is not perfect (and his paper did not suggest it was). Like all order-3 magic hypercubes, it is classed as a simple order-3 magic hypercube..

    The smallest perfect magic tesseract possible is order-16. John Hendricks constructed the first one in 1999, and Clifford Pickover confirmed that it summed correctly to 524,296 in the required 163,840 ways (straight line paths only). [4] [5]

    hh-1  Modern day presentation of planck's 'octahedroid'

    hh-2  This is another aspect of the same tesseract

    [1] Dr. C. Planck, The Theory of Paths Nasik, self-published in Haywards Heath, (England) in November 1905.  18 pages self-cover.
    [2] W. S. Andrews, Magic Squares and Cubes,2nd Edition, Open Court, 1917, pages 363-375 written by C. Planck.
         This book republished by Dover Publ. in 1960. The above fig. 10 and 11 (from the 1905 paper appear in Andrews as fig. 687 and 688.
    [3] A.H. Frost, On the General Properties of Nasik Cubes, Quarterly Journal of Mathematics, 15, 1878, pp 34-49.
    [4] John R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1998,  0-9684700-0-9  pp 126-127 (and private correspondence)
    [5] Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton University Press, 2002, 0-691-07041-5,  page 121.

     

    Reply  Message 6 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:42

    Reply  Message 7 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:44

    How to Draw a 4D Hypercube (My Way)

     
     
    "In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube."
    See all 23 photos
    "In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube."

    A Tesseract is a Four Dimensional Hypercube

    If you're drawing a square on a flat sheet of paper, how many straight lines does it take? Four. If you're drawing a cube, how many squares (sides) does that take? Six. So if you're drawing a tesseract, how many cubes does that take? Eight!

    In this hub, I'm going to show you how to draw your very own tesseract! The lengths of the lines and the angles won't be exact, however, because I'm not using a ruler for this tutorial.

     
    Step 1
    See all 23 photos
    Step 1

    First: How to Draw an Ordinary Cube

    Step 1: Draw two lines of equal length, attempting to keep them an equal space apart, at slightly different heights.

     
    Step 2
    See all 23 photos
    Step 2

    Step 2: Connect the two lines as shown, creating what looks like a smooshed square, or a fat diamond that fell over.

     

     
    Step 3
    See all 23 photos
    Step 3

    Step 3: Draw four parallel lines stemming from each of the shape's four corners.

     

     
    Step 4
    See all 23 photos
    Step 4

    Step 4: Connect the ends of the two top lines, the ends of the two bottom lines, and then connect each bottom line with the line above.

     

     

     

    Visual instructions for drawing the tesseract follow below:

    Step 5
    See all 23 photos
    Step 5
    Step 6
    See all 23 photos
    Step 6
    Step 7
    See all 23 photos
    Step 7
    Step 8
    See all 23 photos
    Step 8
    Step 9
    See all 23 photos
    Step 9
    Step 10
    See all 23 photos
    Step 10
    Step 11
    See all 23 photos
    Step 11
    Step 12
    See all 23 photos
    Step 12
    Step 13
    See all 23 photos
    Step 13
    Step 14
    See all 23 photos
    Step 14
    Step 15
    See all 23 photos
    Step 15
    Step 16
    See all 23 photos
    Step 16
    Step 17
    See all 23 photos
    Step 17
    Step 18
    See all 23 photos
    Step 18
    Step 19
    See all 23 photos
    Step 19
    Step 20
    See all 23 photos
    Step 20
    Step 21
    Step 21
    Step 22
    Step 22
    Step 23
    Step 23

    And there you have it! A complete two dimensionally rendered tesseract, and only twenty-three steps later. I hope you enjoyed my little tutorial!

    If you're interested to learn more about four dimensional geometry, try getting your hands on a copy of Geometry, Relativity, and the Fourth Dimension by Rudolf v. B. Rucker, published in 1977.

     

    Reply  Message 8 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:46

    Hypercube construction

    27 February 2008 at 11:22 am (Mechatronics, Un-Usual Post, Videos)

    I’ve decided to write this post in order to try to give an answer to Eggshell Robotic question “Tesseract / Hypercube – Mechanical Possible?“… The main reason to make a post on my blog rather then making a comment on Eggshell Robotic’s post is mainly the fact that I wrote a lot… and I use images and links… so here it is :

    A hypercube is basically 8 cubes organized using the same logic as we use to build a square (2 dimensions) with strokes (1 dimension), and to build a cube (3 dimensions) with squares (2 dimensions)… We take the first cube, join its 6 faces with one face of 6 other cubes… then we go to the 4th dimension by joining all faces that are next to each other… The 8th cube is used to close the hypervolume…

    It might be easier to explain it comparing to the cube construction… We have 6 squares (2D) we take one to use as center. We join 4 other squares around the one in the center… (still in 2D)… Now we rotate the 4 new squares to the upper dimension in order to link theme (now we are in 3D) but the cube is not closed, therefore we have a 6th square to close the missing face of the cube… Here is a site showing part of the process.

    http://beart.wordpress.com/2008/02/27/hypercube-construction/


    Reply  Message 9 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:49

    Hypercubes

    The most famous example of a higher dimensional object is the hypercube. Broadly speaking, the hypercube is to the cube as the cube is to the square. But, what does this mean? What can we determine about the geometrical properties of the hypercube?

    The best way to grasp a picture of the hypercube is to work up to it dimension by dimension. We can start in zero dimensions and proceed onwards to produce a whole family of cubes. A point does not have any extension. It is the unique zero dimensional object. If we allow our point to sweep out a length of one unit, say along the x axis, we can produce a line segment. Next we can allow the line segment to sweep out a distance of one unit in a perpendicular direction, say one unit along the y axis. The figure that we will produce is a square. Next, our square can sweep out a distance of one unit in a direction perpendicular to the first two directions, say the direction along the z axis. The result will be a unit cube.

    What happens next? We now seem to have run out of perpendicular directions in which we can continue this procedure. However, we can imagine that there are further perpendicular directions and envisage abstract higher dimensional space. There is no reason why we should restrict ourselves to the three dimensions of physical space; as long as we reason clearly and consistently, we can deduce the mathematical properties of higher dimensional objects, even if they cannot be physically realised in our three dimensional universe. We can simply declare that we are now working in four dimensional space and see where our exploration leads us. If we find ourselves in an interesting place, then that will be sufficient reward for our journey. Often the research of pure mathematicians is guided by the search for interesting abstract scenery rather than a quest for scientific utility. In our abstract four-dimensional space, there will be a fourth axis that is perpendicular to our other three axes. We can label this axis the w axis. Measuring the distance parallel to each of the four axes gives us four coordinates that will specify the position of any point within our four dimensional space.

    With a fourth perpendicular direction that we might label the w axis, we can now take the cube and sweep it one unit along this direction. The object that we will generate is the four dimensional equivalent of the cube. It is the object of our dreams, our hypercube. So what does it look like? Is there any way that we can visualize it?

    Our sections through a cube formed a sequence of squares. And similarly, our sections through the hypercube form a sequence of cubes. [We can imagine going down to two and one dimension to see how the analogy works in the case of the two-dimensional cube, which we usually call a square and the one-dimensional cube, which we usually call a line segment. (Our sequence of sections of a square are a sequence of line segments. Our sequence of sections of a line segment is a sequence of points.)]

    What else can we infer from this description of a hypercube? If we draw the hypercube with an exaggerated perspective it will help us to understand the analogy with a cube. First we can draw a transparent cube face-on, so that we can see the back face within the front face and with edges connecting the corners of the front face to the corners of the back face. We can make a similar drawing of a transparent hypercube. If we position the hypercube in the same orientation as in our previous discussion, in our perspective drawing we will see a large cube, with a smaller cube within. Each of the corners of the outer cube are connected by edges to the corners of the inner cube. What we are looking at is the front cube (the outer cube) with the back cube, which is further away in four-dimensional space, appearing within it because its size appears diminished with distance. This is completely analogous to seeing the back face of the transparent cube within its front face.

     
     
       
     
     

    In the projection shown above, the front and back squares of the cube both look square, but the other four square faces of the cube do not look square. Similarly, in the projection of the hypercube, the front and back cubes of the hypercube both look cubical, but the projection has distorted the shapes of the other six cubes that form the hypercube. However, there are more symmetrical ways to project a hypercube down to two dimensions, in which the distortion is shared equally between all eight cubes. Such a projection is shown below. It gives a better feel for the structure of the hypercube. The outlines of each of the eight cubes can be picked out in this projection.

     
     
       
       
     
     
     

    In the illustration shown above right, the axes have been given arbitrary labels: 'w', 'x', 'y' and 'z'. In four-dimensional space these would be the four mutually perpendicular axes. In this projection down to two dimensions, the 'w' and 'y' axes remain perpendicular and the 'x' and 'z' axes remain perpendicular, but the angle between the 'w' axis and the 'x' axis is now 45°.

     
     
         
     
     

    On the left above, the twelve edges of one of the 8 cubic cells of the hypercube have been coloured red. The edges and faces of this cube are aligned with the 'w', 'x' and 'z' axes. In four-dimensional space, this cube is perpendicular to the 'y' axis. On the right, the edges of the opposite cubic cell have been coloured magenta. If we were looking along the 'y' axis, these would correspond to the front and back cube of the hypercube. Note that each corner of one cube is connected to a corner of the opposite cube by a blue edge that is oriented in the 'y' direction, the direction that is perpendicular to the two cubes.

     
     
     

    The image below shows a three-dimensional projection of the edges of a hypercube. The shadow of the hypercube is the two-dimensional projection that we have been considering above. In a sense the three-dimensional figure is a three-dimensional shadow of the four-dimensional hypercube.

     
     
     

    Reply  Message 10 of 25 on the subject 
    From: BARILOCHENSE6999 Sent: 17/03/2013 03:53
  •  

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    Drawing the Hypercube - University of South Alabama

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    Acknowledgement. This talk was supported by the Ministry of. Education Science and Technology (MEST) and the Korean Federation of Science and ...
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    How to Draw a hypercube « Math

     
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    Like Scratches in the Sand - Tesseract Tutorial

     
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    How to draw a 4-D hypercube without having to actually understand the math! In case, I don't know, this is a skill you think you will need someday.
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    Skeptic's Play: Hypercubes and hypercube nets

     
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    04/03/2008 – The hypercube is hard to draw! I often try to draw hypercubes in the margins of my notebooks when I should be listening to lectures (true story!).
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    Hypercube

     
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    What is the Hypercube? Cubes in Perspective Central Projections Nets Cross- Sections, More Drawings The n-dimensional Cube The Hypercube on the Internet ...
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    Hypercube - The Geometer's Sketchpad Resource Center

     
    www.dynamicgeometry.com/.../Hypercube.htmlEn caché - Similares - Traducir esta página
    ... to my friends how a multidimensional object can be visualized in two-space, or what it means to "draw a picture" of a hypercube (a four-dimensional object).
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