This instructable will teach you how to draw a 2-D representation of a X-Dimensional Cube. The formula is both simple and complex at the same time so I guess some would say elegant.
In this example I will draw a tesseract - a 4-D cube, in 2-D
To understand will have to assume some things. The lines you see here have both width and length but in actuality they simply have length, so you will have to assume no width at all. I am using thick lines to make thing a bit easier to see.
Some concepts when seen in 2D:
A 1-Dimensional cube would be represented by a line
A 2-D cube would be a square
A 3-D cube is the standard view we have all seen many times
A 4-D cube something a bit unusual
An X-D cube what you've got to be kidding---not really--it can be done just follow the steps
Step 1: Let's Make a 2D Cube, Drawn in 2D, a Square
In order to make a cube we need another dimension that is at 90 degrees from the first second image
To do this you:
Choose the number of dimensions draw a line to represent a 90 degree angle from the starting point. Each one of these line segments are usually the same length but may appear shorter due to perspective and each is known as a representation.
Now here comes the formula:
Starting at the Startpoint(SP) move to the endpoint of a given segment and the draw a new unused representation that has not been drawn on this line segment previously.
Now this sounds complex so lets go through it
This time to draw 2-D Cube , a Square
A line is drawn since we want a 2-D we need another line segment at a 90 degree angle from the first (a new representation). This is easy as we commonly see this and understand it. so now we have the next image.
Now let's do the horizontal line first, using the formula we go to the SP, move to the end point -EP of that segment(away from the SP) and draw the segment we have not draw on this segment yet which is a vertical line and we have next image.
Repeating the formula
Go to the other segment the vertical one in this case we start at the SP and move to the EP of that segment and draw the UNDRAWN segment which is the horizontal.
Now since all the line segments have all the representations drawn on them the image is done as in the last image and we have a square.
Step 2: Moving on to a 3D Representation, a 3D Cube, Drawn in 2D
We are now going to do a 3d cube which we have seen many times and learned how to draw when we were kids but not this way and now we know why it works.
First we start with the representations of the three dimensions.
Remember these are all line segments that are 90 degrees from each other but shown in perspective.
The first image of the three representations.
Your imagination now has to start coming into play that diagonal line is assumed to be 90 degree off of both the horizontal and the vertical if the image was seen in perspective. It becomes easier to follow if you can do this.
Now we follow the formula.
Starting with the vertical first go to the SP and move to the EP of the ENTIRE segment and we nee to draw any undrawn representations
Completing this step the image looks like the next image. Remember each segment must contain all representations.
Lets go through this one
Starting on the vertical you go to the first EP which is the top of that line here you need to draw the undrawn representations(the horizontal and diagonal)
Now continuing on for the diagonal line you just drew, you have to start at the NEW EP which is the opposite end of the representation just drawn and continue which means to draw the only one not drawn yet, the horizontal.
For the horizontal you do the same process move to the NEW EP which is the opposite end and continue by drawing the only representation not yet drawn on this segment a diagonal.
Lets do the diagonal next go to the SP and move to the EP of that segment and draw one of the two we have not drawn on this segment the next image shows the addition of the vertical representation. Now since this representation actually intercept another existing line segment we can add any additional length as far as the representations go. Since we have a new horizontal line we add on we do not need to duplicate this line segment because that does nothing for us.
The second part of the diagonal again starts at the SP and move to the EP at this point the only representation left that we have not used on this line segment is horizontal this is shown in the next image. Finishing this step up we need to move to the new EP of the segment and draw the remaining representation the vertical, shown in the next image.
The last segment(horizontal) is finished doing the same thing again start at the SP move to the EP of the segment and draw the undrawn segments again when these intermediary segments are drawn they intersect existing line segments you can add those to the current segment so you do not have to draw them again first you will add the new diagonal shown in the next image and then the vertical in the last image for this step.
You now have a Cube drawn in 2-D I am sure we are all familiar with this
Step 3: The Tesseract, a 4D Cube Drawn in 2D
Now to draw the tesseract we do the exact same thing only it is a bit more complex.
I won't type out the details of each step as it will become repetitive, but it is the same formula done over and over until there are no more representations to draw.
The one thing that is a bit different in this and it will hold true for everyone is that you have to imagine the new representations as all being 90 degrees from each other which will be difficult as you are probably not accustomed to thinking like this.
Here is the base image to start from the first image.
Try to image the new diagonal as twisting off in a different direction and from this perspective it could be coming up toward you or anything......happy imagining...
NOW it starts getting interesting... Remember start at the SP go to the EP draw an undrawn representation move to new EP and repeat until all representations exist on the given line segment.
I will try to group the pictures so you can see how the line segments extend
Remember this looks ugly as you draw it by the line segments but you can start to see how things will be tying together
The images here show only one possible pathway to finishing the tesseract. It is up to you just follow the formula
One thing to notice is that if done right at the end of adding to the line segments all the remaining representations for given segment it should end up as a closed image. This is a plus to let you know everything is correct but again this is only one way to do the segments some may choose to draw all the possible representations at a single endpoints repeating this for all the endpoints and being EXTREMELY careful and do not duplicate any.
This will look and go crazy when done this way but it will work.
The next image shows what that could look like
It is looking a bit crazy but it is also coming together the next image shows the remaining representations being added.
The last image is the complete tesseract.
Now looking carefully you can see many versions of our original cube offset from each other which is where our imagination comes in.
This is just the different views of the tesseract
Remember this is a simple formula and can be used to draw an X-D cube
Go to SP move to EP draw an unused representation move to new EP continue on segment until all representations exist on the segment.
Now you can use this formula in the construction of a true tesseract in a 3d modeling software and using a 3d printer create one.
The trick for using the 3d modeling is to build the perspective in by reducing the length of the new diagonal representations so it can be created say by 25%. This way it can be created but you will have to imaging that all segments being the same length when actually looking at it.
For example holding a pencil so it is pointing away from you when you look at it you know it is longer than it looks if the image was a flat picture and measuring it on the picture.
Although this is a 4D Cube the same formula can be expanded to draw a 5D, 6D, 7D ... XD cube I will let you explore the possibilities.
The last image is an actual tesseract drawn in 3D using solidworks and this formula.