What to buy

You can buy many kinds of graphics programs.


The easiest kind of graphics program to use is called a paint program. It lets you easily create pictures on your screen by using a mouse.

How paint programs arose In January 1984, Apple Computer Company began selling the Mac computer. It was the first affordable computer that included a mouse ó and the first affordable computer that included a mouse, and the first affordable computer that included a good paint program.

The Macís paint program was called Mac Paint.

It was invented at Apple Computer Company in 1984 by Bill Atkinson. It ran just on the Mac, was included free with the Mac and showed consumers why a Mac was better than an IBM PC: the Mac let you paint a picture on your screen, and the IBM PC couldnít do that yet.

I explained Mac Paint in the 14th edition of The Secret Guide to Computers. If youíd like that edition, phone me at 617-666-2666.

Mac Paint had one major limitation: it couldnít handle colors. It handled just black-and-white, because the original Mac came with just a black-and-white screen.

(Years later, Apple began charging for Mac Paint, Ann Arbor Software invented an improved version called "Full Paint", and Silicon Beach invented a further improvement called "Super Paint". Modern Macs have color.)

The next major advance was Deluxe Paint.

It was invented in 1985 by Dan Silva in California and published by Electronic Arts. It was much fancier than Mac Paint and performed gorgeous color tricks.

It ran just on Commodoreís Amiga computer. It was why Commodoreís Amiga became popular. Because of Deluxe Paint, the Amiga quickly developed a reputation as the best computer for generating color graphics.

(Years later, the Amiga faced competition, Commodore went bankrupt, and Electronic Arts made versions of Deluxe Paint for the IBM PC and the Apple 2GS. I explain Deluxe Paintís IBM PC version on pages 272-276.)

Windows includes a free a paint program.

In Windows 3.0, 3.1, and 3.11, the free paint program called Paintbrush. Itís a stripped-down version of "PC Paintbrush", which was invented by Z-Soft. Windows 95 includes a free paint program called Paint, which is an improved Paintbrush. I explained Paintbrush on page 112, Paint on pages 101-102.

Paintbrush, PC Paintbrush, and Paint are all worse than Deluxe Paint, except for Deluxe Paintís one glaring problem: Deluxe Paint is a DOS-based program that hasnít been updated to handle Windows and new video cards.

The best paint program for kids is Kid Pix, published by Broderbund.

It runs on all popular computers (IBM, Mac, and others). While you paint, it makes funny sounds and talks to you in both English and Spanish. Besides letting you create your own shapes, it includes lots of fun little pre-drawn shapes (stars, snowflakes, trees, etc.), which you can include in your paintings to create backgrounds and pixie dust.

By using Kid Pix, you can create impressive artwork in just a few seconds! Of all the paint programs you can buy, Kid Pix is the one that give you pleasure fastest! Though the pre-drawn shapes look kid-like, they look like they come from talented kids! Kid Pix is the only program where itís even more fun to erase your work than to create it, since Kid Pix gives you many dramatic ways to get rid of your painting, such as by dynamiting it: boom!

Educators have given Kid Pix many awards for turning kids into creative artists.

The best paint program for professional artists is Painter, published by Fractal Design.

It was designed for the Mac but now also runs on the IBM PC. Painter amazes artists because it makes the computerís screen accurately imitate different kinds of brushes, inks, and other artist tools. You can choose whether to make the screen look like youíre painting in oil, chalk, charcoal, watercolor, or whatever other medium you wish. You can fine-tune each tool, change precisely how "drippy" each tool is, and change the "bumpiness" of the paperís texture.

It even includes a "van Gogh" mode, which lets you paint by using the same kinds of brushstrokes as the artist Vincent van Gogh.

Though Painter can use a mouse, Painter imitates artist tools more accurately if you buy a pressure-sensitive graphics tablet (which comes with a pen that records not just where youíre pressing but also how hard youíre pressing). The most popular pressure-sensitive tablets are made by Wacom, Kurta, Calcomp, and Summagraphics.

Unfortunately, Paint is expensive ($379) and requires a powerful computer. If you have less than a Pentium (or Power Mac) with less than 16 megabytes of RAM, Paint runs too slowly to be enjoyable.

Fractal Design sells a stripped-down version, called Dabbler, for just $49. It also sells a black-and-white version called Sketcher, which runs faster since it doesnít have to deal with colors.

Since Paint, Dabbler, and Sketcher are intended just for creative artists who like to draw squiggles, they donít contain commands to draw geometric shapes. For example, they donít contain commands to draw a oval, circle, rectangle, or square. All other popular paint programs include such commands.

How paint programs work Each paint program considers your screen to made of thousands of pixels (dots). The paint program remembers the color of each pixel. The colors of all the pixels are stored in RAM while youíre painting. You need lots of RAM if your screen is large & has many pixels, or if you insist on using lots of colors.

All paint programs suffer from the following problem:

If you use a paint program to create a shape, then try to shrink that shape, then change your mind and try to expand the shape back to its original size, the final result looks crude and lacks the details that were in the original.

Thatís because a paint program shrinks a shape by using fewer pixels, and so some of the pixels that contained details are discarded. The lack of detail becomes noticeable when you try to expand the shape back to its original size.

Another problem is that when you try to rotate a shape, the shape looks cruder, because the shapeís pixels get slightly misplaced by "round-off error". If you try to rotate a shape several times, the pixels get progressively more misplaced, and the shape looks cruder and cruder.

When trying to paint, if you expand or twirl,

You get a result that makes you want to hurl.

Paint programs are called bitmapped graphics programs.


A draw program does not store the color of each pixel. Instead, a draw program stores a memo about a geometric shape and the color of the entire shape.

For example, a draw program stores a line by storing just its starting point, angle, length, and color; it stores a circle by storing just the circleís center, radius, and color. By contrast, a paint program would consume lots of RAM storing the color of each of the thousand of pixels that are on the line or circle.

Draw programs are also called vector-based graphics programs.

A draw program works faster and more accurately than a paint program if youíre drawing geometric shapes. A draw program has no problem handling expansions and rotations. But it has difficulty handling squiggles, since it tries to view each squiggle as made up of many tiny arcs.

A draw program lets you name different objects, put them in front of other objects, then later move the objects to reveal objects that were hidden. Most paint programs canít do that: in a paint program, creating a new shape automatically erases any shape that was underneath ó except for a crude feature that lets you have two "layers": a "background" and a "foreground".

In a draw program, you can point to an object you drew and change its color, thickness, or style. In a paint program, the only way to change the appearance of what you drew is to draw it over again.

Unfortunately, the typical draw program is confusing to use, because when you look at whatís on your screen youíre not sure which "objects" the stuff youíre seeing is part of.

The first popular draw program was Mac Draw, which ran on the Mac. Now most draw programs use Windows instead.

The most popular serious draw programs are Windows Draw (by Micrografx), Micrografx Designer (which is a beefed-up version of Windows Draw), and Corel Draw (which comes from a Canadian Company called Corel).

Each modern Windows word-processing program also includes a stripped-down draw program, free:

To draw while using Microsoft Word, click the Drawing button.

To draw while using Word Perfect, choose "Draw" from the Graphics menu.

To draw while using Word Pro, choose "Drawing" from the Create menu.

To draw while using Microsoft Worksí word processor, choose "Drawing" from the Insert menu.


You can buy a program that does computer-aided drafting & design (CAD). Such a program resembles a draw program but does more math.

For example, it can print mock blueprints, with the lengths of all parts marked. It can even compute the surface area (square feet) of any shape, so you can compute how much material to buy to build your structure and cover it.

It lets you give fancy geometric commands, such as "draw a 37-degree angle, but make the point be round instead of sharp, so nobody gets hurt" or "draw a circular arc that goes through these three points" or "draw a line that grazes these two circles, so itís tangent to them".

The most famous CAD program is Autocad. Thatís what most architects and engineers use. Itís published by Autodesk and very expensive: about $3,000! Cheaper CAD programs that cost under $50 are for use at home, to help Joe Six-Pack design his backyard deck and interior designers plan purple bathrooms.

Most CAD programs include pre-built shapes that you can put in your drawings.

The pre-built shapes are exactly the right size and shape to represent toilets, sinks, stoves, and other household fixtures. Each shape is called a symbol. You can buy a bunch of extra symbols; each bunch is called a symbol library.

Photo manipulation

To put photographs into your computer, use a scanner (which scans in sheets of paper), or a digital camera, or a digital video camera (such as the Connectix Quick Cam), or Snappy (which attaches your computer to a camcorder, TV, or VCR).

Once the picture is in your computer, you can manipulate it by using a paint program. Better yet, use a program that specializes in the fine art of manipulating photographs.

The best photo-manipulation programs are Adobeís Photoshop, Adobeís Photo Deluxe (which is cheaper), Corelís Photo-Paint, Microsoftís Picture It, and a fun program by Metatools called Kaiís Power Goo (which lets you stretch and shrink each part of a personís face, to create weird facial expressions).

Other graphics software

To spice up your word-processing documents, you can buy clip art, which consists of funny little cartoonish illustrations. Modern spreadsheet program (such as Excel) can create pie charts, line graphs, and bar charts. Presentation programs (such as Microsoftís Power Point and Lotusís Freelance) create slide shows. The best way to combine graphics with word processing is to get a desktop-publishing program, explained on pages 286-287.

Deluxe Paint

To create beautiful pictures on your computerís screen, get a program called Deluxe Paint ó or one of its cousins.

Deluxe Paint was invented in California by Dan Silva in 1985. Then he invented an improved version (Deluxe Paint 2). Then he developed a further improvement (Deluxe Paint 3), which included advanced graphics tricks and also animation. His programs all ran on the Commodore Amiga. They were published by Electronic Arts.

An Electronic Arts employee ó Brent Iverson ó developed an IBM version of Deluxe Paint 2. Then another Electronic Arts employee ó Steve Shaw ó developed an IBM version of Deluxe Paint 3 and called it Deluxe Paint Animation. He also invented a lesser version called Deluxe Paint 2 Enhanced, which includes all the features of Deluxe Paint 3 except animation.

Electronic Arts also developed an Apple 2GS version of Deluxe Paint 2.

Electronic Arts also sells several paint programs for the Mac: Studio 1 has animation but no color; Studio 8 and Studio 32 have color but no animation. They were all developed by an independent group of French programmers inspired by Dan Silva.

In this chapter, Iíll concentrate on the IBM versions. Each IBM version requires 640K of RAM and a mouse.

Choose an IBM version

Which IBM version should you buy ó Deluxe Paint 2, Deluxe Paint 2 Enhanced (which should be called Deluxe Paint 2Ĺ), or Deluxe Paint Animation (which should be called Deluxe Paint 3)?

Intelligence Deluxe Paint Animation understands more commands than the other versions, and itís the only version that produces animation.

Price Deluxe Paint 2 has been included free with many computers and software packages. Thatís because itís considered obsolete! Discount dealers sell Deluxe Paint 2 Enhanced and Deluxe Paint Animation for about $100 each.

Hard disk Deluxe Paint 2 Enhanced and Deluxe Paint Animation require a hard disk. If you donít have a hard disk, you must buy Deluxe Paint 2.

VGA Deluxe Paint Animation requires a VGA (or MCGA) video card. If your video card is worse than that ó if you have just EGA, CGA, or Hercules ó you must buy Deluxe Paint 2 or Deluxe Paint 2 Enhanced.

High resolution Unfortunately, Deluxe Paint Animation operates just in low resolution (320-by-200). (Instead of using RAM to store higher resolution, it uses RAM to store your animation.) If you insist on full VGA resolution (640-by-480), buy Deluxe Paint 2 or Deluxe Paint 2 Enhanced. To operate in Super VGA resolution (800-by-600 or 1024-by-768), you must buy Deluxe Paint 2 Enhanced.

Summary Deluxe Paint 2 is obsolete. Use it only if you get it free or youíre broke or you lack a hard disk.

If your computer has a poor video card (Hercules monochrome, CGA, or EGA), get Deluxe Paint 2 Enhanced.

If your computer has a good video card (MCGA, VGA, or Super VGA), get either Deluxe Paint Animation or Deluxe Paint 2 Enhanced. Deluxe Paint Animation offers animation, but Deluxe Paint 2 Enhanced offers higher resolution.

Assumption In the rest of this section, I assume youíre using Deluxe Paint 2 Enhanced or Deluxe Paint Animation.

Copy to the hard disk

When you buy Deluxe Paint 2 Enhanced or Deluxe Paint Animation, you get four 5ľ-inch floppies and two 3Ĺ-inch floppies.

Hereís how to copy the 5ľ-inch floppies to your hard disk. (Copying the 3Ĺ-inch floppies is similar.)

Turn on the computer without any floppy in drive A. When you see the C prompt, put the PROGRAM Disk into drive A and type "a:". The computer will display an A prompt.

Type "install". The computer will say "Installation".

If youíre using Deluxe Paint Animation, press ENTER twice. If youíre using Deluxe Paint 2 Enhanced instead, press ENTER once, then type "dp" (so the screen says "dp" instead of "DPAINT"), then press ENTER again.

When the computer tells you, put the other three disks in drive A and press ENTER.

The computer will say, "Installation Complete!" (Then if youíre using Deluxe Paint 2 Enhanced, do this: type "century", press ENTER, and wait for the computer to copy fonts to the hard disk.)

Run the paint program

To start using the paint program, turn on the computer without any floppy in drive A.

To do Deluxe Paint Animation, type "do da". To do Deluxe Paint 2 Enhanced, type "do dp".

(That "do" method works if you put the DO.BAT file onto your hard disk as I recommended in the MS-DOS chapter. If you have not put DO.BAT onto your hard disk, do Deluxe Paint Animation by typing "cd da" and then "da"; do Deluxe Paint 2 Enhanced by typing "cd dp" and then "dp".)

Choose a video mode

If youíre using Deluxe Paint 2 Enhanced, you must choose a video mode. (If youíre using Deluxe Paint Animation, you donít have to choose a video mode, and you can skip ahead to the next section.)

Deluxe Paint 2 Enhanced shows you this list of video modes:

a. CGA 320 x 200 4 colors

b. CGA 640 x 200 2 colors

c. EGA 320 x 200 16 colors

d. EGA 640 x 200 16 colors

e. EGA 640 x 350 16 colors

f. MCGA 320 x 200 256 colors

g. MCGA 640 x 480 2 colors

h. VGA 320 x 200 16 colors

i. VGA 640 x 200 16 colors

j. VGA 640 x 350 16 colors

k. VGA 640 x 480 16 colors

l. Hercules 720 x 348 2 colors

m. Tandy 320 x 200 16 colors

o. Amstrad 640 x 200 16 colors

p. Extended VGA 640 x 400 256 colors

q. Extended VGA 640 x 480 256 colors

r. Extended VGA 800 x 600 2 colors

s. Extended VGA 800 x 600 16 colors

t. Extended VGA 800 x 600 256 colors

u. Extended VGA 1024 x 768 2 colors

v. Extended VGA 1024 x 768 16 colors

Type a letter from "a" to "v". Hereís which letter to type:

Video you bought Type this letter

Hercules monochrome "l"

CGA "a" for lots of colors, "b" for high resolution

EGA "e"

MCGA "f" for lots of colors, "g" for high resolution

VGA "f" for lots of colors, "k" for high resolution

A plain VGA system is limited to 640-by-480 resolution and has a 256K video RAM. If your VGA system can handle higher resolution or has extra video RAM, you have extended VGA, which lets you type these letters instead:

Extended VGA you got

Resolution Video RAM Type this letter

640-by-480 512K "q" for lots of colors, "k" for high resolution

800-by-600 256K "f" for lots of colors, "s" for high resolution

800-by-600 512K "t"

1024-by-768 256K "f" for lots of colors, "u" for high resolution

1024-by-768 512K "t" for lots of colors, "v" for high resolution

If you choose a letter near the end of the alphabet ("p" through "v"), you face two complications:

The computer might say, "Specify which card is active". To reply, press the SPACE bar, look at the menu of VGA card manufacturers, and type your manufacturerís code letter. (If your VGA card is manufactured by a company thatís not on the menu, try pretending that your VGA card is an AST VGA Plus, which handles video modes q and r.)

The computer might say, "Not enough memory". That means your computer doesnít have enough expanded RAM. Choose a different letter instead.

Move the mouse

Look at the computerís mouse. The mouseís tail is a cable that runs from the mouse to the computer. The area where the tail meets the mouse is called the mouseís ass.

The mouseís underside ó its belly ó has a hole in it, and a ball in the hole.

Put the mouse on your desktop and directly in front of you. Make the mouse lie flat (so its ball rubs against the desk). Make the mouse face you so you donít see its ass.

In the middle of the screen, youíll see a cross, which is called the mouse pointer.

Move the mouse across your desk. As you move the mouse, remember to keep it flat and facing you. As you move the mouse, the cross moves also.

If you move the mouse to the left, the cross moves to the left. If you move the mouse to the right, the cross moves tot he right. If you move the mouse toward you, the cross moves down. If you move the mouse away from you, the cross moves up.

Practice moving the cross by moving the mouse. Remember to keep the mouse facing you at all times.

If you want to move the cross far and your desk is small, move the mouse until it reaches the deskís edge; then lift the mouse off the desk, lay the mouse gently on the middle of the desk, and rub the mouse across the desk in the same direction as before.

Draw a squiggle

On the top of the mouse, youíll see 2 or 3 rectangular buttons you can press. The main button is the one on the left.

The middle of the screen is a white, rectangular area thatís huge: it consumes most of the screen. That area is called the painting area. Itís where you draw your pictures.

Try this experiment. Put the cross in the middle of the screen, in the middle of the painting area. While holding down the mouseís left button, move the mouse. That activity ó moving the mouse while holding down the mouseís left button ó is called dragging. As you drag, youíll be drawing a squiggle.

For example, try drawing a smile. To do that, put the cross where you want the smile to begin (at the smileís upper-left corner), the depress the mouseís left button while you draw the smile. When you finish drawing the smile, lift the mouseís button. Then draw the rest of the face!

Click an icon

At the screenís right edge, you see these pictures:

One-Dot Brush Squiggle

Line Arc

Filled Rectangle Filled Circle

Filled Polygon Filled Blob

Fill Spray

Brush Pickup Text

Grid Perspective

Magnify Hand

Color Pickup Symmetry

Undo Clear

Gradient Pattern

Color Indicator



Color Palette




Those pictures are called icons (or tools).

Try this experiment: move the mouse pointer (the cross) to those icons. When the cross reaches those icons, the cross turns into an arrow.

Line Hereís how to draw a line thatís perfectly straight.

Move the mouse pointer until it turns into an arrow, and the arrowís tip is in the middle of the Line icon. Then tap the mouseís left button. (Thatís called clicking the Line icon.)

Then put the mouse pointer in the middle of the screen, in the middle of the painting area, where you want the line to begin. Drag to where you want the line to end (by moving there while holding down the mouseís left button).

The line you desired will appear!

Practice drawing lines. When you finish drawing lines and want to draw squiggles instead, click the Squiggle icon. (Since the Squiggle icon looks like a paintbrush, itís also called the Freehand Brush icon).

Practice! Draw lines, then squiggles, then lines, then squiggles. When you feel comfortable, explore the following icons, which are more advanced.Ö

Arc To draw an arc (a smooth curve), click the Arc icon (which is also called the Curve icon). Then put the mouse pointer in the painting area, where you want the arc to begin, and drag to where you want the arc to end. Lift your finger off the mouseís button, move the mouse until the arc has the curvature you wish, then click.

Undo If you make a mistake, click the Undo icon. That makes the computer erase the last object you drew (or undo your last activity). The screen will look the same as before you drew that object (or did that activity).

Clear To erase everything you drew on the screen (so the screen becomes white and you can start over), click the Clear icon (which says CLR). If youíre using Deluxe Paint Animation, you must then press ENTER.

Color Palette Normally, the computer draws black shapes (on a white background). To draw in a color other than black, look at the Color Palette icon (which is divided into many colors), and click the color you want. The color you choose is called the foreground color.

For example, to draw a blue object, do this: click blue, then draw the object. To erase part of an object, do this: click white, then cover the object by drawing with white paint.

If youíre using Deluxe Paint 2 Enhanced and chose a 2-color video mode, the only colors in the palette are black and white.

If youíre using Deluxe Paint Animation (or Deluxe Paint 2 Enhanced in a 256-color mode), the Color Palette icon shows just some of the colors. To see other colors, repeatedly click the Right Arrow icon (at the screenís bottom right corner). To go back to the colors you were seeing before, click the Left Arrow icon. When you see your favorite color, click it.

Color indicator In the middle of the Color Indicator icon, you see the color that you picked to be the foreground color.

Color pickup Suppose youíve drawn a picture, and one of the colors youíve used is so nice that you want to use it again. Click the Color Pickup icon (which looks like an eyedropper that soaks up ink), then click your pictureís middle, where you used that color. That color will become the foreground color.

Filled rectangle To draw a rectangle whose sides are perfectly straight, and whose middle is filled in, click the Filled Rectangle icon. Then put the mouse pointer in the painting area, where you want the rectangleís top left corner to be. Drag to where you want the rectangleís opposite corner.

Filled circle To draw a circle whose middle is filled in, click the Filled Circle icon. Then put the mouse pointer in the painting area, where you want the circleís center to be. Drag until the circle is as big as you wish.

Filled polygon To draw a polygon whose middle is filled in, click the Filled Polygon icon. Then click in the painting area, where you want the polygonís first corner to be.

Click where you want the polygonís second corner to be. Presto ó the computer draws a line from the first corner to the second corner!

Click where you want the polygonís third corner to be. The computer draws a line from the second corner to the third corner.

Keep clicking, until youíve clicked all the corners. Then press the SPACE bar, which makes the computer draw a line from the last corner back to the first corner and fill in the polygon.

Filled blob To draw a blob (lumpy circle) whose middle is filled in, click the Filled Blob icon (which is also called the Filled Freehand Shape icon). Then put the mouse pointer in the painting area, where you want the blob to begin. Drag in a circular motion (so youíre drawing a lumpy circle). When you lift your finger from the mouseís button, the computer completes the circular shape (by drawing a line back to where you began) and fills in the middle.

Fill Suppose your drawing contains a big area thatís all the same color. For example, suppose you drew a big filled rectangle, circle, polygon, or blob. Hereís how to change the color of that area.

Click in the Color Palette, at the new color you want, so it becomes the foreground color. (The new color can be red, blue, black, white, or any other color you wish.)

Click the Fill icon (which looks like a paint bucket).

The mouse pointer turns into a paint bucket. Move the mouse pointer carefully, so the tip of the paint (spilling out of the bucket) is in the middle of the large area whose color you want to change. Then click.

The entire areaís color will change to the foreground color.

If you make a mistake, click the Undo icon and try again.

You can use that technique to change the color of any big area thatís all the same color. For example, if you draw a big yellow sun, you can change the sunís color to orange.

Hereís how to start drawing a landscape:

Clear the screen (by clicking the Clear icon). Then draw the horizon, as follows: by using the Squiggle icon, draw a horizontal black squiggle all the way across the screen. Make sure the squiggle goes all the way across the screen ó from the screenís far left to the screenís far right ó so that there are no gaps in the horizon, and so that itís impossible to travel from the sky (above the horizon) to the ground (below the horizon) without crossing the horizon.

Then make the sky blue. To do that, click the blue part of the Color Palette icon, then click the Fill icon, then click in the middle of the sky. (If youíre using Deluxe Paint 2 Enhanced in a 2-color mode, pick black instead of blue and make a night scene.)

The entire sky will turn blue (or black).

If you made a mistake and forgot to make the horizon go all the way across the screen, the skyís blue paint will leak through the horizon and make the ground blue. Click the Undo icon and try again!

To make the ground be desert brown or lawn green or fantasy-land purple, click your favorite ground color in the Color Palette, then click Fill, then click in the middle of the ground.

Hereís how to draw a face thatís embarrassed:

Clear the screen (by clicking the Clear icon). Begin using the Squiggle icon, draw the faceís black outline (a circle with two bumps on it ó for the ears). Make sure the faceís bumpy circle is a complete circle and has no gaps. Make sure itís impossible to go from the middle of the face to the screenís edge without crossing the faceís circle.

Then make the face turn red. To do that, click the red part of the Color Palette icon, then click the Fill icon, then click in the middle of the face. The entire face will turn red.

If you made a mistake and forgot to make the faceís circle be complete, the faceís red paint will bleed through the gap in the faceís circle and make the rest of the screen be bloody. Yuk! Click the Undo icon and try again!

Spray To vandalize your own drawing by using a can of spray paint, click the Spray icon (which looks like a can of spray paint and is also called the Airbrush icon). Then put the mouse pointer in the painting area, where you want to begin spraying, and drag!

Text To put words in the middle of your picture, click the Text icon. Then click in the painting area, where you want the words to begin. Type the words.

Grid To draw simple diagrams more easily, click the Grid icon before you draw. Then anytime you click or drag the mouse (except when drawing squiggles), the computer will automatically nudge the mouse pointer so that its X and Y coordinates are a multiple of 8.

For example, try this experiment. Click the Grid icon. Draw a line (by using the Line icon). Now try to draw another line that starts at exactly the same point as the first line. Because of the grid, you donít have to bother putting the mouse pointer exactly where the first line began; just put the mouse pointer close to that point, and the computer will automatically nudge the mouse pointer so that its coordinates are a multiple of 8 ó and exactly where the first line began.

Magnify To magnify part of the screen temporarily, so you can see more clearly what youíre drawing there, click the Magnify icon (which looks like a magnifying glass). Then click in the part of painting area you want to magnify.

The screen splits into two parts. The left part of the screen shows the objects in their actual size; the right part of the screen shows the objects magnified.

Go ahead: continue drawing! Whatever you draw will appear in the left part of the screen (actual size) and simultaneously in the right part of the screen (magnified).

When you finish needing magnification, click the Magnify icon again. The screen will return to its usual, unmagnified state.

Hand While looking at a magnified part of the screen, hereís how to nudge the magnifying glass so you see a slightly different view.

Click the Hand icon. (Itís also called the Grabber icon.) Put the mouse pointer in the middle of the magnified view. Drag a short distance in the direction that you want to nudge the view.

Symmetry To create perfect symmetry, click the Symmetry icon (which looks like a snowflake or the image seen through a kaleidoscope). Then move the mouse pointer near the center of the painting area. Youíll see the mouse pointer and 11 clones of it ó 12 pointers altogether, arranged in a circle.

Go ahead: continue drawing! Whatever you draw will be duplicated 11 extra times, so youíll see 12 copies of your drawing. The 12 copies are arranged in a circle and rotated, and 6 of those copies are mirror images (flipped backwards). Groovy!

When you finish symmetric drawing, click the Symmetry icon again. Then youíll have just one mouse pointer, instead of 12.

Pop out a menu

You can make the screen display extra icons. Hereís how.

Zigzag Try this experiment. Point at the Line icon, then hold down the mouseís left button awhile. A new menu pops out onto the screen. (Itís called a pop-out menu). The menu shows two icons. One of them is a copy of the Line icon. The other is the Zigzag icon: itís a line with a bend in it. To use the Zigzag icon, drag to it.

Hereís how to draw a zigzag (try it!). Drag to the Zigzag icon (which pops out of the Line icon). Then click in the painting area, where you want the zigzag to begin. Click where you want the zigzagís first bend. Click where you want the zigzagís second bend. Click at each additional bend. Click where you want the zigzag to end. Then press the SPACE bar. (If the zigzagís ending point is the same as where the zigzag began, pressing the SPACE bar is optional.)

The Zigzag icon is also called the "Connected Lines" icon or "Polyline" icon.

After youíve used the Zigzag icon, that icon stays on the screen, where the Line icon used to be.

If you ever want to draw a simple line again, make the Line icon reappear. Hereís how. Point at the Zigzag icon. Hold down the mouseís left button. Youíll see the pop-out menu again. Drag to the pop-out menuís Line icon.

Practice drawing zigzags, then simple lines, then zigzags again. When you feel comfortable, explore the following icons, which are more advanced.Ö

Zigzag arcs To draw a zigzag made of a series of arcs instead of lines, drag to the Zigzag Arcs icon, which pops out of the Arc icon. Then point in the painting area, where you want the first arc to begin, and drag to where you want that arc to end. Lift your finger off the mouseís button. Move the mouse until that arc has the curvature you wish, then click.

Click where you want the second arc to end, and click where you want that second arc to curve. Click where you want the third arc to end, and click where you want the third arc to curve. Do the other arcs.

When you finish the last arc, press the SPACE bar.

The Zigzag Arcs icon is also called the "Connected curves" icon or "Polycurve" icon.

Extended color palette If youíre using Deluxe Paint Animation (or Deluxe Paint 2 Enhanced in a 256-color mode), try this trick. In the Color Palette icon, point at one of the colors, and hold down the mouseís left button awhile. Youíll see a pop-out menu displaying all 256 colors simultaneously. Drag to the color you want. It becomes the foreground color.

If youíre using Deluxe Paint 2 Enhanced in a 16-color mode, try this similar trick instead. In the Color Palette icon, point at your favorite color, and hold down the mouseís left button awhile. Youíll see a pop-out menu that displays 16 modifications of that color. (The computer creates the modifications by adding tiny polka dots made of the other colors. The modifications made by polka dots are called dithers.) Drag to whichever dither you like. That dither becomes the foreground color.

Modified rectangles Point at the Filled Rectangle icon, and hold down the mouseís left button.

Youíll see four pop-out icons. One of them is a copy of the Filled Rectangle icon. Another (Filled Square) acts like the Filled Rectangle icon but always produces a perfect square. The other two icons (Unfilled Rectangle and Unfilled Square) let you draw an outline shape whose middle is not filled in.

Modified circles Point at the Filled Circle icon, and hold down the mouseís left button.

Youíll see six pop-out icons. One of them is a copy of the Filled Circle icon. The other pop-out icons are Filled Ellipse, Filled Rotated Ellipse, Unfilled Circle, Unfilled Ellipse, and Unfilled Rotated Ellipse.

To draw a filled ellipse (oval), choose the Filled Ellipse icon. Then in the painting area, imagine a rectangle just big enough to contain the ellipse you want. (Thatís called the bounding rectangle.) Point at that rectangleís top left corner, and drag to the opposite corner. The ellipse will appear.

To draw a filled rotated ellipse, choose the Filled Rotated Ellipse icon. Then click in the painting area, where you want the ellipseís center to be. Move the mouse until the ellipse is as fat and tall as you wish. Then drag in a circular motion, until the ellipse is rotated to the angle you wish.

Unfilled polygon The Unfilled Polygon icon pops out of the Filled Polygon icon.

Modified squiggles Point at the Squiggle icon, which looks like a paintbrush thatís drawing a squiggle. Hold down the mouseís left button. Youíll see three pop-out icons.

One of them is a copy of the Squiggle icon.

Another popped-out icon (Dotted Squiggle) acts like the Squiggle icon but makes the squiggle be a series of dots instead of a continuous curve. If you want the dots to be far apart, drag the mouse fast while drawing the dotted squiggle. That icon is also called the Discontinuous Freehand Brush icon.

The other popped-out icon (Stopped Squiggle) produces just a single dot instead of a complete squiggle. Since it stamps just one dot onto your screen, itís also called the Single-Stamp Freehand Brush icon.

Magnification level When you use the Magnify icon, it normally magnifies objects by a factor of 4, so that each object appears 4 times as wide and 4 times as tall. Thatís called a magnification level of 4x.

To choose a different magnification level, point at the Magnify icon and hold down the mouseís left button. Youíll see a pop-out menu that offers these magnification levels: 2x, 3x, 4x, 6x, 8x, 12x, and 16x. Drag to the magnification level you want.

Choose a brush

To draw without a computer, you can buy a collection of paintbrushes. To draw a thick line, use a thick brush. To draw a thin line, use a brush thatís tiny and fine. To create a weird texture, paint with a toothbrush. To create a big weird texture, paint with a hairbrush. To create a gigantic weird texture, get together with your friends, undress, cover yourselves with paint, and roll around on a large sheet of canvas. (Thatís called Technicolor mud wrestling. Itís a quick, fun way to create an abstract pop mural. The mural makes a great conversation piece when you match the shapes on it with the various people and their parts.)

Your computer lets you paint on the screen by using all kinds of brushes. You can even invent your own brush, having any shape you like!

Hereís how to pick a brush.

Built-in brushes Point at the One-Dot Brush icon, and hold down the mouseís left button.

Youíll see 18 brush shapes. Some of them are round brushes. Some are square. Some have multiple tufts (like a tiny toothbrush, or like a paintbrush whose bristles have been purposely frayed apart). Some are chisel-pointed (like a miniature ax, for calligraphy and Japanese effects). Theyíre called the 18 built-in brushes.

Drag to whichever built-in brush interests you, and try it! The computer will use that brush to draw all shapes (squiggles, lines, arcs, etc.), until you switch to a different brush.

Custom brushes Hereís how to invent your own brush.

First, decide what shape you want the brush to produce when you tap the brush onto paper. Do you want the resulting shape to be a tiny dot, or a circle, or a square, or an irregular blob, or blob thatís the same shape as a ear, or a blob that looks like Bart Simpsonís hair, or some other weird shape?

Draw that shape in the painting area. (For example, if you want the brushís blob to look like Bart Simpsonís hair, draw a picture of Bart Simpsonís hair.) When drawing that shape, use your favorite colors!

To make that shape become your brush, click the Brush Pickup icon. Then draw a rectangle around the shape: point where you want the rectangleís top left corner, and drag to where you want the rectangeís opposite corner. When you lift your finger off the mouseís button, the shape becomes your new brush. Its colors become the new colors youíll be drawing in.

The computer will automatically click the Dotted Squiggle icon for you. Go have fun! Drag the mouse wherever you wish! As you drag, youíll be drawing dotted squiggles by using your new brush and colors.

The brush you invented is called a custom brush. It will be your brush until you invent a different custom brush or switch back to one of the 18 built-in brushes (by clicking the One-Dot Brush icon).

Classic computer art

During the 1960ís, many creative ideas were generated about how computers would someday create their own weird art, using a wild combination of formulas and random numbers, and unshackled by the bounds of human culture.

Hereís how to make the computer produce wild art, by using the wonderful classic tricks invented in the 1960ís and 1970ís.Ö

Gray levels

You can express every black-and-white photograph as a table of numbers. Each number in the table represents the darkness of a different point ó the higher the number, the darker the point. The "darkness numbers" are called gray levels. To feed a picture into the computer, type in the table of gray levels. Or aim a special camera at the object you want pictured; the camera system will automatically compute the gray levels and send them to the computer via a wire.

You can program the computer to change the gray levels in any weird way you wish, and draw the result. In the 1960ís, the Computer Technique Group of Japan did this to an ordinary photograph of John Kennedy:

Shot Kennedy Diffused Kennedy








Kennedy in a Dog


Hereís what the group did to a photograph of Marilyn Monroe:

Monroe in the Net




















Csuri & Shaffer fed the computer a realistic line drawing of an old man; hereís what came out:

Random Light and Shadow

I did this with the help of a computer:







































The Pin-Up has these specifications.Ö

scene: a scantily clad woman sitting on a stool

4 gray levels

4 symbols (1 for each gray level: a blank, a period, an asterisk, and a dollar sign)

1537 symbols altogether (53 rows ◊ 29 columns)

In the specification, the numbers are small, yet the picture is clear. To obtain the clarity, I did non-computerized finagling.

At Bell Telephone Laboratories, Knowlton & Harmon produced a picture with much larger specifications.Ö

scene: two sea gulls flying in the clouds

16 gray levels

141 symbols (each gray level has several symbols; the computer chooses among them at random)

11616 symbols altogether (88 rows ◊ 132 columns)

Instead of using blanks, periods, asterisks, and $, they used cats, battleships, swastikas, and other weird shapes. Here are the 141, listed from lightest to darkest, with some repetitions:































The picture is several feet long. Seen from a distance, it looks like this:


Hereís a close-up view of part of one of the gullís wings:









If you donít like sea gulls, how about Mona Lisa?

In 1971, Michael Hord made the computer turn photographs into artistic sketches. Hereís what the computer did to a photograph of his boss, and to a photograph of a colleagueís girlfriend:

















































To draw each sketch, the computerís camera scanned the original photograph and found the points where the photograph changed dramatically from light to dark. Then, on a sheet of paper, it plotted those points; and through each of those points, it drew a short line perpendicular to the direction in which the original photograph darkened.

More precisely, hereís what the computer did.Ö It looked at four adjacent points on the original photograph:



It computed the darkness of each of those points. Then it computed the "darkening in the X direction", defined as:

(darkness at B) + (darkness at D) - (darkness at A) - (darkness at C)

Then it computed the "darkening in the Y direction", defined as:

(darkness at A) + (darkness at B) - (darkness at C) - (darkness at D)

Then it computed the "overall darkening", defined as:

(darkening in the X direction)≤ + (darkening in the Y direction)≤

If the overall darkening there turned out to be large, the computer sketched a short line, in the vicinity of the points ABCD, and perpendicular to the direction of darkening. More precisely, the lineís length was 1, and the lineís slope was:

the darkening in the X direction

the darkening in the Y direction


Hereís how to make an L slowly become a V. Notice that the letters L and V are both made by connecting three points:



Let 1" be the point halfway between 1 and 1'; let 2" be halfway between 2 and 2'; and let 3" be halfway between 3 and 3'. Then 1", 2", and 3" form a shape thatís halfway between an L and a V:



The process can be extended further:


Turning one shape into another (such as turning an L into a V) is called a metamorphosis or morphing. The intermediate shapes (that are between the L and the V) are called the morphs.

Using that method, the Computer Technique Group of Japan gradually turned a running man into a Coke bottle, and then into Africa:

Running Cola is Africa

The group turned this head into a square:

Return to a Square












The head on the left returns to a square by using arithmetic progression: the lines are equally spaced. The one on the right uses geometric progression instead: the lines are close together near the inside square, but further apart as they expand outward.

Csuri & Shaffer exploded a hummingbird:

Chaos to Order









The hummingbird at the far right was obtained from the one at the far left, by moving each line a random distance and in a random direction (between 45į and -45į).

Computers can make movies. The best movie ever made by a computer is called Hunger (or La Faim). It was made under the auspices of the Canadian Film Board. Itís a 10-minute cartoon, in color, with music; but it goes far beyond anything ever done by Walt Disney. It uses the same technique as Running Cola is Africa: it shows objects turning into other objects.

It begins by showing a harried, thin executive at his desk, which has two phones. One of the phones rings. He answers it. While heís talking on that phone, his other phone rings. To talk on both phones simultaneously, his body splits in two. (How does a single body become two bodies? By using the same technique as turning a running man into a coke bottle.)

On the other side of his desk is an armchair, which turns into a secretary, whose head turns into a clock saying 5PM, which tells the executive to go home. So he stretches his arms in front of him, and becomes his car: his hands become the headlights, his arms become the front fenders, his face becomes the windshield. You have to see it to believe it.

He drives to a restaurant and gets the waitress, who turns into an ice-cream cone. Then he eats her.

As the film progresses, he becomes increasingly fat, lustful, slothful, and miserable. In the end, he falls into hell, where heís encircled by all the poor starving naked children of the world, who eat his flesh. then the film ends. (Donít see it before eating dinner!)

It combines computer art and left-wing humanitarian politics, to create an unforgettable message.

Now morphing is being applied to color photographs and video images. For example, Hollywood movies use morphing to show a person gradually turning into a monster; environmentalists use morphing to show a human baby gradually turning into a spotted owl; and portrait photographers who have gone high-tech use morphing to show you gradually turning into the person you admire most (such as your movie idol or your lover).

Order versus disorder

Computer artists are starting to believe that art is a tension between order and disorder. Too much order, or too much disorder, will bore you. For example, in Chaos to Order, the hummingbird on the left is too orderly to be art. The hummingbird on the right is more interesting.

Consider Gulls. Seen from a distance, itís an orderly picture of gulls. Seen up close, itís an orderly picture of a cat or battleship or swastika. But from a middling distance, it looks like disorderly wallpaper: the symbols repeat, but not in any obvious cycle. That element of disorder is what makes the picture interesting.

At first glance, Pin-Up is just a disorderly array of periods, asterisks, and dollar signs. At second glance, you see order: a girl. Art is the formation of order from disorder.

A first glance at Monroe in the Net shows order: a piece of graph paper. A second glance shows disorder: some of the graphís lines are inexplicably bent. A third glance shows order: Marilyn Monroeís face pops out at you. Her orderly face is formed from the disorder of bent lines.

Return to a Square uses arithmetic progression and geometric progression to create an over-all sense of order, but the basic elements are disorderly: a head thatís bumpy, and a panorama of weird shapes that lie uncomfortably between being heads and squares but are neither.

Many programs create disorder by random numbers. Chaos to Order uses random numbers to explode the hummingbird. Gulls uses random numbers to help choose among the 141 symbols.

An amazing example of random numbers is this picture by Julesz & Bosche:


















To your eyes, the picture seems quite ordered. Actually, itís quite disordered. One pie-shaped eighth of it is entirely random; the other seven eighths are copies of it. The copying is the only element of order, but very powerful. Try this experiment: cover seven-eighths of the picture. Youíll see that the remaining eighth is totally disordered, hence boring.

That program imitates a childís kaleidoscope. Do you remember your childhood days, when you played with your kaleidoscope? It was a cardboard "telescope" that contained a disorganized pile of colored glass and stones, plus a series of mirrors that produced eight-way symmetry, so that what you saw resembled a giant multicolored snowflake. The program by Julesz & Bosche uses the same technique, computerized. Hundreds of programmers have imitated Julesz & Bosche, so now you can buy kaleidoscope programs for the IBM PC, Mac, Apple 2, Radio Shack TRS-80, and all other popular computers. Or try writing your own!

Take this test:








































One of those is a famous painting (Composition with Lines, by Piet Mondrian, 1917). The other was done by a computer (programmed by A. Michael Noll in 1965). Which one was done by the computer? Which one do you like best?

The solution is on the next page, but donít peek until youíve answered!

The computer did the top one.

The programmer surveyed 100 people. Most of them (59) thought the computer did the bottom one. Most of them (72) preferred the top one ó the one that was actually done by the computer.

The test shows that people canít distinguish computer art from human art, and that the computerís art is more pleasing that the art of a famous painter.

The computerís version is more disordered than Mondrianís. The computer created the disorder by using random numbers. The survey shows that most people like disorder: Mondrianís work is too ordered. It also shows that most people mistakenly think the "computer" means "order".


Try this experiment. On a piece of paper, put two dots, like this:


The dots represent little insects, or "bugs". The first bug is looking at the second bug. Draw the first bugís line of sight:


Make the first bug take a step toward the second bug:


Make the second bug run away, in any direction:



Now repeat the entire process. Again, bug 1 looks at bug 2; draw its line of sight:



Bug 1 moves toward bug 2:



Bug 2 keeps running away:




If you repeat the process many times, you get this:








The "motion of bug 1" looks like a curve. (In fact, itís a parabola.) The "curve" is composed of many straight lines ó the lines of sight. Thatís how to draw a fancy curve by using straight lines.

Each straight line is called a tangent of the curve. The entire collection of straight lines is called the curveís envelope. Creating a curve, by drawing the curveís envelope, is called stitching the curve ó because the lines of sight act as threads, to produce a beautiful curved fabric.

You can program the computer to draw those straight lines. Thatís how to make the computer draw a fancy curve ó even if you know nothing about "equations of curves".

To get a curve thatís more interesting, try these experiments:

What if bug 2 doesnít walk in a straight line?

What if bug 2 walks in a curve instead?

What if bug 1 goes slower than bug 2, and takes smaller steps?

What if the bugs accelerate, or slow down?

What if there are three bugs?

What if bug 1 chases bug 2, while bug 2 chases bug 3, while bug 3 chases bug 1?

What if there are many bugs?

What if they all chase each other, and their starting positions are random?

What if there are just two bugs, but the bugs are Volkswagens,

which must drive on a highway having nasty curves?

Show the bugs driving on the curved highway.

Their lines of sight are still straight;

but instead of moving along their lines of sight,

they must move along the curve that represents the highway.
















What if each bug has its own highway, and all the bugs stare at each other?

Here are some elaborate examples.Ö

Four bugs chase each other:

The next example, called Compelling, appeared in the famous book and movie, The Dot and the Line. (Norton Juster made it by modifying art that had appeared in Scripta Mathematica.) It resembles the previous example but makes the 4 bugs start as a rectangle (instead of a square), and makes the bug in the top left corner chase the bug in the opposite corner (while looking at a nearby bug instead).

















Enigmatic (from The Dot and the Line) makes 3 bugs chase each other, while a fourth bug stays motionless in the center:

















I invented Kite, which makes 8 bugs chase each other:

I also invented Sails, which makes 14 bugs chase each other:



















Elliptic Motion (by my student Toby DíOench) makes 3 bugs stare at each other, while they travel on 3 elliptical highways:






















Archimedean Spiral (by Norton Starr) puts bugs on circles. The bugs stare at each other but donít move:


A fractal is an infinitely bumpy line. Hereís how to draw one.

Start by drawing a 1-inch line segment:


In the middle of that segment, put a bump and dip, like this:




Altogether, that bent path is 2 inches long. In other words, if the path were made of string, and you stretched the string until it was straight, the string would be 2 inches long. Thatís twice as long as the 1-inch line segment we started with. So hereís the rule: putting a bump and dip in a path makes the path twice as long.

That bent path consists of seven segments. Put a bump and a dip in the middle of each segment, like this:





Altogether, those bumps and dips make the path twice as long again, so now the path is 4 inches long.

Again, put a bump and dip in the middle of each segment, so you get this:





Again the pathís length has been doubled, so now the path is 8 inches long.

If you again put a bump and dip in the middle of each segment, the pathís length doubles again, so the path becomes 16 inches long. If you repeat the procedure again, the path reaches 32 inches.

If you repeat that procedure infinitely often, youíll develop a path thatís infinitely wiggly and infinitely long. That path is longer than any finite line segment. Itís longer than any finite 1-dimensional object. But it still isnít a 2-dimensional object, since it isnít an "enclosed area". Since itís bigger than 1-dimensional but isnít quite 2-dimensional, itís called 1Ĺ-dimensional. Since 1Ĺ contains a fraction, itís called fractional-dimensional or, more briefly, fractal.

Look out your window at the horizon. What do you see? The horizon is a horizontal line with bumps (which represent hills and buildings and other objects). But on each hill, you see tiny bumps, which are trees; and on each tree, you see even tinier bumps, which are leaves; and on each leaf, you see even tinier bumps, which are the various parts of the leaf; and each part of the leaf is made of even smaller bumps (molecules), which have even smaller bumps (atoms), which have even smaller bumps (subatomic particles). Yes, the horizon is an infinitely bumpy line, a fractal!

You can buy software that creates fractals. Computer artists use fractal software to draw horizons, landscapes, and other bumpy biological objects. For example, they used fractal software to create landscapes for the Star Wars movies. You can also use fractals to draw a bumpy face that has zillions of zits.

Now you understand the computer artistís philosophy of life: "Lifeís a lot of lumps."

Whatís art?

To create art, write a weird program whose consequences you donít fully understand, tell the computer to obey it, and look at the computerís drawing. If the drawing looks nice, keep it and call it "art" ó even if the drawing wasnít what you expected. Maybe it resulted from an error, but so what? Anything interesting is art.

If the drawing "has potential" but isnít totally satisfying, change a few lines of the program and see what happens ó or run the program again unchanged and hope the random numbers will fall differently. The last thing to invent is the title. Whatever the drawing reminds you of becomes the title.

For example, thatís how I produced Kite and Sails. I did not say to myself, "I want to draw a kite and sails". I just let the computer pick random starting points for the bugs and watched what happened. I said to myself, "Gee whiz, those drawings remind me of a kite and sails." So I named them Kite and Sails, and pretended that I chose those shapes on purpose.

That method may seem a long way from DaVinci, but itís how most computer art gets created. The rationale is: donít overplan.Ö let the computer "do its own thing"; it will give you art that escapes from the bounds of human culture and so expands your horizons!

Modern style

Computer art has changed. The classic style ó which youíve been looking at ó consists of hundreds of thin lines in mathematical patterns, drawn on paper and with little regard for color. The modern style uses big blobs and streaks of color, flashed on a T.V. tube or film, which is then photographed.

Uncreative art

Youíve seen that computers can create their own weird art by using a wild combination of formulas and random numbers, unshackled by the bounds of human culture.

Today, programs such as Deluxe Paint let people use computers to create art easily and cheaply. Unfortunately, the typical person who buys a graphics program uses it to create the same kind of junk art that would be created by hand ó just faster and more precisely. Thatís the problem with computers: they make the production of mediocrity even easier and more glitzy.

3-D drawing

The computer drew these three-dimensional surfaces:

Three Peaks

by John Szabo

















Those were done for the sake of art. This was done for the sake of science:

Population Density in the U.S.

by the Harvard University Mapping Service














The hardest part about three-dimensional drawing is figuring out which lines the computer should not show, because theyíre hidden behind other surfaces.


Try this experiment. Put your finger on the bridge of your nose (between your eyes). Now move your finger 2 inches to the right (so that your finger is close to your right eye). Then move your finger 3 inches up (so that your finger is near the upper right corner of your forehead). From there, move your finger 8 inches forward (so that your finger is 8 inches in front of your forehead).

Your fingerís current position is called (2,3,8), because you reached it by moving 2 inches right, then 3 inches up, then 8 inches forward. The 2 is called the X coordinate; the 3 is called the Y coordinate; the 8 is called the Z coordinate.

You can reach any point in the universe by the same method! Start at the bridge of your nose, and get to the point by moving right (or left), then up (or down), then forward (or back).

The distance you move to the right is called the X coordinate (if you move to the left instead, the X coordinate is a negative number). The distance you move up is called the Y coordinate (if you move down instead, the Y coordinate is a negative number). The distance you move forward is called the Z coordinate (if you move back instead, the Z coordinate is a negative number).

Projecting the coordinates

To draw a picture of a three-dimensional object, put the object in front of you, and then follow these instructions.Ö Pick a point on the object. (If the object has corners, pick one of the corners.) Figure out that pointís X, Y, and Z coordinates (by putting your finger on the bridge of your nose and then seeing how far you must move your finger right, up, and forward to reach the object).

Then compute the pointís projected X coordinate (which is X/Z) and the pointís projected Y coordinate (which is Y/Z). For example, if X is 2 and Y is 3 and Z is 8, the projected X coordinate is 2/8 (which is .25) and the projected Y coordinate is 3/8 (which is .375). On graph paper, plot the projected X coordinate and the projected Y coordinate, like this:






Then plot the point:






Go through the same procedure for every point on the object (or at least for the corners). Connect the dots and ó presto! ó you have a three-dimensional picture of the object! And the picture is mathematically accurate! Itís what artists call a "true perspective drawing".

To make the picture look traditionally beautiful, place the object slightly to the left of you and slightly below your eye level, so that all the X and Y coordinates become negative.

Computerizing the process

You can program the computer so that if you input a pointís X coordinate, Y coordinate, and Z coordinate, the computer will calculate the projected X coordinate (from dividing X by Z) and the projected Y coordinate (from dividing Y by Z) and plot the point on the computerís screen (by using high-resolution graphics).

The easiest way to draw three-dimensional pictures is to buy a special three-dimensional arm that attaches to an Apple computer. To draw a picture of an object, move the mechanical arm until the armís finger touches the object. Immediately the armís software computes the X coordinate, Y coordinate, and Z coordinate of the touched point; you donít need a ruler! It also computes the projected X coordinate and the projected Y coordinate and plots the points on your television. If you have a graphics printer, the software also plots the point on your printerís paper.