Archives May 2021

Random Sine Wave Flowers Made With turtleSpaces Logo

This simple yet attention-grabbing Logo procedure is a real visual treat, creating random three-dimensional ‘flowers’ based on the sine function. This could spice up a math class or just provide a brief introduction to sine waves in general.

The procedure makes use of the move primitive, which allows the turtle to move based on an arbitrary number of degrees away from its current heading, on the X and Y planes. When combined with the sine function, it produces these interesting ‘natural’ 3D designs.

Live model view – click and drag to rotate
TO randomflower
  clearscreen randompencolor
  ;clear the screen and select a random
  ;pen color
  dountil and 200 < abs (:x - :y) 200 < abs (:y - :x) [
    make "x random 2000 make "y random 2000
  ;set :x and :y containers to a random value between 0 and 2000
  ;BUT make sure they aren't within 200 of each other.
  ;The results in that range aren't aesthetically pleasing.
  repeat 3600 [
    ;we move down a tenth of a degree each iteration, so to form
    ;a complete 'circle' we need 3600 repetitions.
    move 0.5 {:x * sin repcount :y * sin repcount}
    ;the move primitive takes a number of turtle steps as its
    ;first argument, and then either a single degree value (X)
    ;or a list of two values (X and Y), which turn the turtle
    ;the specified number of degrees on each plane.
    ;In the case of this example, we're passing a multiple of
    ;:x and :y based on the sine of the current interation
    ;(repcount). This produces the 'flower' effect.
    down 0.1

You could put a wait into the loop so that students can see the process a bit more clearly.

Don’t forget that you can click and drag the view to orbit around the model, click with both buttons and drag to ‘pan’ the orbit point, and use the mouse wheel (or right-click and drag up and down) to zoom in and out.

Children can pick their favourites and share them with the class!

This is the flower listing as seen in the turtleSpaces application editor

PS Adding the following two lines into the repeat 3600 loop can add a little ‘pizzaz’ in the form of frags, ‘fragments’ (think of a broken mirror) or triangles made of three arbitrary points. The frag primitive creates a triangle from the last three points the turtle has stopped at, while pin and pinfrag drop ‘pins’ at points you choose and generate triangles from the last three of those pin points respectively. When combined with the sine flower procedure they add a bit of ornamentation. Just remember to setfillcolor!

    if 0 = remainder repcount 10 [pin] 
    if 0 = remainder repcount 30 [pinfrag]

Thanks for reading! Please let us know how you go at integrating turtleSpaces into your classroom’s studies. E-mail us at 

Simple Hexagon Pattern

Here is a simple Logo project to get started with. It creates a colorful filled hexagon pattern using the polyspot primitive.

The hexagon pattern zoomed out

TO hex
  clearscreen penup
  ;not drawing lines
  repeat 10 [
    ;do all of the following ten times
    setpos [-149 -270]
    ;sets the 2D turtle position (on the current Z plane)
    ;you can use setposition to set all three co-ordinates
    forward repcount * 34
    ;positions the turtle to start the line
    ;you could use some math with setpos to accomplish this
    ;choose a random fill color
    polyspot 20 6
    ;make a filled polygon of size 20, with 6 sides
    repeat 10 [
      ;make ten more polygons
      slideright 30
      forward 17
      ;randfc is the short form of randomfillcolor
      polyspot 20 6]
    ;return to the start of the loop

A turtleSpaces (app) screenshot of the editor

The hexagon pattern viewed from a different 3D angle

Past and Future Turtles: The Evolution of the Logo Programming Language (Part 1)

When I was a young child, one of my best friends was a turtle.

Not a real turtle, although that would have been fun, but a virtual one. My turtle lived inside of an Apple II, an early 8-bit computer, at my elementary school.

When I first met her, my turtle, she didn’t do much. She just sat there. It didn’t take long for me to realize that in order to get her to do something, I would need to prompt her. To do this, there was a literal prompt on the screen, beckoning me to type something.


And so I typed FORWARD, thinking that this should, obviously, move the turtle forward.


Undeterred, I revised my input to FORWARD 10, just picking an arbitrary number. Although I was only six years old, I deduced that FORWARD needed a quantity, quantities were numbers and well, ten was a good start.

And hooray, the turtle moved! I was in love with my turtle from that moment on. I learned that she could also move BACK, and turn LEFT and RIGHT. I didn’t know what degrees were when I first met my turtle, but by the time I finished our first session together, I did!

Seymour Papert in 1971

That afternoon in 1981, I had learned some of a programming language called Logo. The father of Logo was a mathematician and philosopher named Seymour Papert. Interested in learning how to improve mathematics education in young children, in the early 1960s, he had studied under Jean Piaget, a Swiss psychologist who championed a philosophy of learning known as constructivism.

When children are given new information, for example, that a full turn is made up of 360 degrees, they are likely to only accept that information if they already have the underlying understanding required, such as that a turn can be divided in half, quarters and so on, down to 360ths. Or even the concept of a number as high as 360 can exist in the first place.

Otherwise, they are likely to either misinterpret the new information to fit their current understanding (“I only understand numbers as high as 100, and so they must mean 100”) or will simply ignore it, with a silent internal ‘syntax error’.

This can improve if a) the teacher explains numbers up to 360, b) the teacher explains division and then finally c) the teacher explains that when you spin around, you are making a circle (hold out your arm! The point of your finger draws a circle when you spin) and if you spin around halfway, you’re moving 180 degrees and so on.

This may still not be enough. If the child fails to understand a, b and / or c, they still won’t understand it.

However, if the child is able to actively explore these concepts in an environment that has some measure of feedback, the likelihood of their gaining understanding improves. For example, if the teacher suggests they turn randomly, and the teacher tells them how far they have turned each time they do it, they will soon learn that if they turn halfway, they have turned 180 degrees. They will learn that a full turn is 360 degrees. They will understand that the next number after 100 is 101, and hence the next number after 200 is 201. And so on.

This is what Papert learned under Piaget. But obviously, while desirable it’s not practical for each child to have an adult giving them feedback on everything they do until they understand it. The world just doesn’t (and probably will never have) a 1:1 student to teacher ratio. This was a problem Piaget and Papert simply couldn’t surmount.

But in the early 1960s, a new technology was emerging, one that Papert realized could remove his roadblock – the computer.

Computers did not tire, they did not lose patience. Although at that time large and small in number, it was not hard for Papert to see the future, a future where schools were full of terminals connected to computers, one for each child, with which the students could explore ‘mathland’, a simple environment where mathematical concepts could be demonstrated through trial and error. The more input students provided, and the more output they absorbed, the sooner they would grasp concepts. Computers seemed a perfect fit for the task of educating children using constructivist principles.

In 1961, Papert had met an American, Marvin Minsky, at a conference in England. They presented papers that were surprisingly similar, and connected as a result. So, powered by his idea of using computers to teach children, in 1964, Papert went to the US, and joined Minskey’s Artificial Intelligence Group, a team of researchers at the Massachusetts Institute of Technology (MIT).

Papert was impressed with the technology-centric culture at MIT, and attitude that everyone had the potential to learn anything they wanted to, a position Papert shared. The students under Minsky and Papert had free rein to work on whatever they wanted, encouraging experimentation and the practical testing of even the most wild theories. Papert saw how his students, when encouraged, would learn great amounts of knowledge on their own, and collaborate with others to get past roadblocks they encountered in their pursuits. At MIT these students were known as ‘hackers’.

One of these hackers, Daniel Bobrow, graduated and got a job at a research and development (R&D) firm called Bolt, Beranek and Newman. In the mid-1960s they had started exploring computer technologies, and their potential. Bobrow became head of their new Artificial Intelligence group. There, he met Wally Feurzeig, who was leading BBN’s education group. The space race had encouraged the US government to find ways of improving math education in schools, and BBN was hoping to find a solution. They had already developed ‘time-sharing’ technology which allowed a mainframe computer to be used by many people at the same time, they just needed software to run on it that children could use to learn.

Bobrow brought in Papert and introduced him to Feurzeig, and the three discussed the idea of developing a computing language for children, Papert’s Mathland. Around that time, BBN had a visitor who demonstrated a new programming language called BASIC. Rather than using memory addresses, BASIC used line numbers to track execution, and named variables that transparently linked to memory addresses. These abstractions made it much easier for non-computer scientists to create computer programs.

BBN’s researchers created a version of BASIC they called Telcomp, which Feurzeig modified to include strings, and called Stringcomp. BBN trialed Stringcomp (using teletypes, typewriters connected to a remote computer) in eight elementary and junior high-school classrooms in 1966. That year, Minsky’s assistant at MIT, Cynthia Solomon, joined Feurzeig’s team at MIT.

Seymour Papert at Muzzey Junior High School in 1966

After Papert visited these classrooms and saw the interaction between children and Stringcomp, he became convinced that BASIC was not the answer, and that a new language would need to be written, one more suited to learning.

Importantly, the new language would need to more adequately demonstrate how the computer came to its conclusions. Papert felt that due to computers’ requirements for literal instruction, children would be able to see how the computer ‘thought’ and realize that was how they themselves also thought, helping them break down problems into smaller elements the way a computer would when it solved them. Thus, the new language would serve a dual purpose, both teaching mathematics, and how mathematics are solved.

BASIC was too monolithic in its command-based structure, and those ‘smaller parts’ could not be shown to its users, happening behind the scenes and out of sight.

And so, the new language would both need to be able to accomplish what one could accomplish in BASIC, while doing so using smaller steps that could be inspected by novice programmers. But to avoid needless and redundant repetition of code, these steps would need to be able to be reused. Because of the philosophy behind it, Wally Feurzeig christened the new language ‘Logo’, derived from Logos, the greek word for ‘thought’.

Minsky, Papert and Solomon had a number of discussions about how Logo would work. Solomon had learned some of the Lisp programming language (a language where all code and data are written in the form of expressions and lists, the name Lisp short for LISt Processing) while Minsky’s assistant, and she appreciated its power for manipulating English language strings, which she recognized could be used as a method by which students could observe the effects of computer code on recognizable data, or data that they created themselves.

It also probably didn’t hurt that Lisp had been invented by the same BBN employee (and MIT alumnus) that had developed its time-sharing system. And so, it was decided that the new language would be a variant of Lisp.

However, Lisp had some drawbacks in terms of its use with younger coders. It had an extreme reliance on structure, using an intricate array of parentheses which often led to errors that were hard to resolve (leading to nicknames Lost in Stupid Brackets and Lots of Irritating Superfluous Parentheses). Its interpreter was also bulky, and not interactive, like BASIC.

In designing Logo, Papert, Minsky and Solomon took the expression, list and string elements of Lisp, removing much of the need for brackets by making assumptions about syntax parsing and inputs, and later allowing for the ‘stacking’ of what they called ‘primitives’ (functions which may or may not resolve to a value) on a single line, such that you could type:


and the computer would output 8, or:


and the computer would output 123.

Lists allowed for the string manipulation Solomon had been hoping for:






Logo was a lot more BASIC-like in its syntax and usage, while allowing for many features BASIC did not, such as the ability to create ‘procedures’, lists of primitives that could be named, and executed just like any other primitive, such as:


   REPEAT 4 [
      FORWARD 10
      RIGHT 90


which could be executed simply by typing:


but even better:


      RIGHT 360 / :SIDES


the colon indicating a variable or container for storing a value.

POLY 4 10

would re-create our box, but the user could provide any desired values, like SUM. This satisfied Papert’s need to have both the ability to demonstrate computer problem-solving in smaller chunks, while also allowing for the reuse of code. Students could create the BOX or POLY procedures themselves, then use them in other procedures. A picture would form in their minds of the hierarchy of execution, and they would understand better how the computer, and they themselves, thought.

Except that we are getting ahead of ourselves with BOX and POLY, because in the beginning Logo did not have a turtle! When Logo was initially trialled in a seventh-grade classroom (in 1968 at Muzzey Junior High School in Lexington, Massachusetts), students were using the same teletypes the Stringcomp trial had used, and they used Logo to manipulate lists and strings rather than a turtle. Which worked quite well to engage students who were strong in English, but failed to engage those who were not.

Cynthia Solomon (left) and Seymour Papert (right) at Muzzey Junior High in 1966

Also, on the teletype math was still just numbers, regardless of the more elegant way Logo could break down mathematical equations. Kids who didn’t ‘get’ numbers still didn’t get numbers! Papert craved some sort of graphical representation, but the technology at the time strongly precluded that. And so, towards the end of the trial, he had the idea of creating a physical robot, one that could move about on the floor, and draw out graphical representations of mathematic output.

Papert and Solomon left BBN and headed back to MIT, where they formed the Logo Group. In England, two robots had been developed that could wander about a space, their inventor dubbing them ‘tortoises’. The Logo Group developed their own robots, calling them turtles. Having access to expensive display hardware, they also developed a virtual turtle, one that lived on the computer screen.

They added primitives to Logo to control the turtle: FORWARD, BACK LEFT, RIGHT. And they discovered that Papert had been right – children took to the turtle, the turtle allowing Logo users to have a perspective inside the computer, that of the turtle, rather than depending on an understanding of abstract concepts of code paths or numbers. The procedural structure of Logo meant the turtle could be ‘taught’: taught how to draw a square, taught how to draw a polygon, creating them inside of its environment, interacting it and shaping it like a child scrawling on the wall of their bedroom.

The turtle allowed for an empathetic bond to form between it and the child, giving the child a dual sense of accomplishment when the turtle succeeded at what the child told it to do – both because the child correctly instructed the turtle, and the turtle successfully carried out those instructions, the result of the latter the evidence of the former. The child could also ‘think like the turtle’ when developing procedures and debugging, moving about the room as the turtle would, figuring out roughly what they needed the turtle to do, and then refining their program as needed.

An early Logo video display

While Logo’s string and list facilities are impressive, it is fair to say the turtle made Logo. Both the physical and virtual turtles were used by fifth graders at the Bridge School in Lexington in a trial during 1970 and 1971. But both robots and video terminals were expensive in the 1970s, and schools hadn’t widely adopted computer terminals; as such, there was no widespread use of Logo in the 1970s.

However, the emergence of the home computer market in the late 1970s provided an opportunity for Logo to reach a wider segment of the population. In 1980, Minskey, Papert, Solomon and two Canadians founded Logo Computer Systems Inc. (LCSI) and  in 1981 they published a version of Logo for the Apple II computer, which I booted up and met Seymour, Cynthia and Marvin’s turtle for the first time.

They would have been very happy I learned about degrees (and other things) from their turtle!

But where did Logo go from there? Tune in next time!

In the meantime, try our weblogo… or read the Hacker News discussion!

Example: My Gosh, It’s Full of Stars!

Open in turtleSpaces IDE
Run starscape to create a galaxy of randomly-shaped and colored stars.

TO star :radius :points :size :filled
  ;takes number number number boolean (true or false)
  ;eg star 10 9 40 false
  ;or star 20 5 50 true
  pu dropanchor tether
  ;pull up the turtle pen, drop the 'anchor' to set the 'anchor point' to the current position, and do not change it if the turtle's rotation changes (tether)
  ;the anchor point is the point the orbit primitives orbit around
  if :filled [polyspot :radius :points * 2
    if oddp :points [rt 360 / :points / 4]]
  ;if creating a filled star, create a polygon to fill in the center
  ;if the star has an odd number of points, we need to turn a little
  pullout :radius
  ;pull out from the anchor point, which is currently the turtle's position
  repeat :points [
    ;repeat once for each point:
    pu make "position position
    ;set the :position container to the current turtle position
    orbitleft 360 / :points / 2
    ;orbit around the anchor point (in this case the center) to the left
    ;a fraction of 360 degrees divided by the number of points divided by 2
    pullout :size
    ;pull the turtle out from the center of the star (anchor point)
    setvectors direction :position
    ;point the turtle toward the previous position
    if not :filled [pd line distance position :position]
    ;if not making a filled star, create a line in front of the turtle of
    ;the length required for its end to be the previous turtle position.
    if :filled [frag]
    ;if creating a filled star, create a 'fragment' based on the last three
    ;turtle positions
    pu make "position position
    ;set the :position container to the current position
    pullin :size
    ;pull the turtle in toward the center of the star (anchor point)
    orbitleft 360 / :points / 2
    ;orbit to the left again a similar amound to the last orbitleft
    setvectors direction :position
    ;point toward the previous position
    if not :filled [pd line distance position :position]
    ;create a line if not a filled star similarly to the previous line
    if :filled [frag]
    ;create a fragment if a filled star similarly to the previous frag

TO starscape
  reset ht snappy:setposition [0 0 0] snappy:dropanchor repeat 200 [pu home randvec fd 1000 + random 1000 up 90 snappy:setvectors direction myrtle:position randfc randpc randps randfs star 10 + random 100 3 + random 20 10 + random 90 randbool]

Example: It’s Turtles All The Way Down!

In turtleSpaces, you can ‘stamp’ the current model, leaving a copy of it in the current position. This can be useful in some circumstances, to create artworks made out of more sophisticated models. In the case of this example, we’re going to use the default ‘myrtle’ turtle model to create a descending spiral of turtles.

If you change the colors in the default palette using the definecolor primitive, then those colors are also used to render the turtle model, and by extension the stamp. And so, if we ‘cycle’ the colors, shifting their RGB values from one palette slot to the next, we can create a rotating series of differently-colored turtles.

Note: For reasons, colors 0 (black) and 15 (white) cannot be redefined. But there are 1-14 and 16-63 to play with, which is plenty!

For aesthetic reasons, we want a certain turtle to have the default colors, so we need to add an offset to some of the calculations to make that happen. Finally, we need to position Snappy (the camera or view turtle) in a certain position to get the view we want. Because this involved a fair amount of manual fiddling, I’ve just inserted the position and vectors into the listing to reproduce Snappy’s position and orientation.

Finally, to make the spiral we’re going to use the orbit commands, which rotate the turtle around an ‘anchor point’. The tether primitive allows us to turn the turtle away from the anchor point, which would normally move the anchor point but does not if tether is called. We descend a little each time we move, creating a downward spiral.

TO turtlesallthewaydown

  snappy:setposition [-0.0567188571145 -41.2097873969 -9.14256453631]
  snappy:setvectors [
    [0.0177688320245 0.192937058466 -0.981050233209]
    [-0.00350577343795 0.981211134942 0.192905205264]
    [0.999835975629 1.16396598173E-05 0.0181113694522]
  ;position Snappy where we want him to reproduce the image
  make "defaultcols colors
  ;make a copy of the default color palette
  penup dropanchor
  pullout 50 tether
  ;we want 'orbit' in a circle
  left 90 orbitright 39
  ;because we tethered, we can point away
  ;then orbit around the anchor point
  repeat 800 [
    setfillshade -5 + repcount / 20
    ;set the shade of the fill color, the color used
    ;to create the voxels that make up the turtle, amongst others
    make "offset remainder repcount + 11 14
    ;why +11? So we get the default turtle colors in the right place!
    repeat 14 [
      definecolor repcount item 2 + remainder (:offset + repcount) 14 :defaultcols
      ;shuffle the color palette. We need to start at item 2 because 1 is black
      ;(item is 1-based, whereas definecolor is 0-based -- 0 being black)
    stamp "myrtle
    ;make a copy of myrtle in the current position
    orbitleft 22.95 lower 1


If you replace stamp “myrtle with stamp “astronaut, and before it add a setmodelscale 0.5, like this:

setmodelscale 0.5 stamp "astronaut

you get:

If you change the “astronaut in stamp “astronaut to stamp pick [astronaut spaceship plane] like so:

setmodelscale 0.5 stamp pick [astronaut spaceship plane]

you’ll get: