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Turtle Geometry: The Computer as a Medium for Exploring Mathematics (Artificial Intelligence)

Harold Abelson, Andrea Disessa · 8 HN comments
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Amazon Summary
Turtle Geometry presents an innovative program of mathematical discovery that demonstrates how the effective use of personal computers can profoundly change the nature of a student's contact with mathematics. Using this book and a few simple computer programs, students can explore the properties of space by following an imaginary turtle across the screen. The concept of turtle geometry grew out of the Logo Group at MIT. Directed by Seymour Papert, author of Mindstorms, this group has done extensive work with preschool children, high school students and university undergraduates.
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Aug 06, 2022 · bmitc on Byte Magazine: LISP (1979)
> "The LOGO system supports two different (by no means disjoint) environments: the Turtle, Graphics and Musicbox world (ie: peripheral devices which are controlled by a command language) and the LISP world."

Whelp, the lack of an Oxford comma there really through me for a loop until I continued reading. Anyway...

> "Our experiences, especially with young students, indicate that programming in LOGO may serve as a bridge between natural language communication and reasoning and the formal and abstract symbols and reasoning in mathematics and programming languages."

If anyone is interested in this, there are the following books:

* Exploring Language with Logo:

* Visual Modeling with Logo: A Structured Approach to Seeing:

* Turtle Geometry: The Computer as a Medium for Exploring Mathematics:

* Computer Science Logo Style:

It's disappointing to me that Logo died out and didn't remain around or evolve, and I find it a bit sad that many kids are getting introduced to programming via something like Python or Scratch. I feel there's still space for Logo, and it would be cool to see an easily downloadable and installable version of it.

Wasn't it predictable, though?

Take a look at that issue of BYTE entirely dedicated to LOGO that was referenced multiple times in this discussion.

A dozen articles about the language, with listings. The screenshots? All the same, showing recursive pictures of rectangles and circles. Great. LOGO can do that. But what else? Big empty void there.

Contrast that with what BASIC could do at the time... Is it any wonder LOGO died out and BASIC thrived?

What could Basic do over Logo? As a few of the books I posted show, Logo is a very capable language, ignoring turtles. Was Logo in the 80s somehow less capable?
> Big empty void

See my comment upthread about Turtle Geometry.

> Contrast that with what BASIC could do at the time... Is it any wonder LOGO died out and BASIC thrived?

AIUI, implementations of BASIC on home computers were a lot simpler and more straightforward than LOGO or LISP - for instance, GC in BASIC was an afterthought and only applied to strings. The real competitor to BASIC back then was FORTH.

Mmmh... no.

I grew up and learned programming in that era, bought magazines, typed pages and pages of listings.

98% of the listings in these magazines were BASIC, 1% were assembly, and the rest was... well, others.

LOGO was a niche language then, and Forth even more so.

> It's disappointing to me that Logo died out and didn't remain around or evolve

But it did remain around and evolve.

See, e. g.:

StarLogo Nova:


Scratch was produced at MIT just like Logo and I think it is an evolution of what they were trying to do with Logo.
Despite the heritage of the creators, I personally view Scratch as a misstep. I am a big fan of visual programming, but Scratch is essentially an everyday imperative language with a structured editor and sprites. It doesn't maintain any of the simplicity or elegance of Logo or Lisp, and I'm honestly a bit surprised that the MIT Media Lab invests so much in it.
Turtle Geometry is still worth checking out. The value is in working through the ideas, and details of Logo take up very little of the book.

(I've only glanced through CS Logo Style and haven't seen the other two.)

I think the same basically goes for any of those books. One could easily port the code to Racket, for example.
Yeah. I recommend it because it goes way beyond the stuff another commenter complained about in this thread:

> A dozen articles about the language, with listings. The screenshots? All the same, showing recursive pictures of rectangles and circles. Great. LOGO can do that. But what else? Big empty void there.

E.g. the last chapter is an intro to general relativity, with a simulator for motion in curved spacetime.

(CS Logo Style also covers many topics, but it looked like they were all familiar to me as an experienced programmer. I haven't seen another book for programmers about most of the math in Turtle Geometry.)

* How to Code: Simple Data and How to Code: Complex Data on edX. Taught by Gregor Kiczales, of Common Lisp and CLOS fame. Uses Racket and graphical programs to teach.

* The From Nand to Teris project, The Elements of Computing Systems: Building a Modern Computer from First Principles book, and/or Coursera course. Builds a hardware stack for a CPU and then a software stack (assembler, VM, high-level language).

* The How to Design Programs book. What the edX course above is based upon.

* Structure and Interpretation of Computer Programs (SICP). Uses Scheme. One can use Racket with the `#lang sicp` language.

YouTube playlist of the course by the authors:

* Thinking as Computation: A First Course. Uses Prolog to solve problems of thinking.

* Turtle Geometry: The Computer as a Medium for Exploring Mathematics (shares an author with SICP). Uses Logo to explore turtle geometry/graphics. Can use any modern Logo implementation.

* Starting Forth. Uses Forth.

* Learning Processing: A Beginner's Guide to Programming Images, Animation, and Interaction and also The Nature of Code: Simulating Natural Systems with Processing. Uses Processing and p5.js (the JavaScript version of Processing).

The author's YouTube channel:

The book Turtle Geometry [1] by Harold Abelson of SICP fame is a pretty cool exploration of mathematics all the way up to non-euclidean geometry using turtle graphics.


My favorite Logo-for-relative-grownups work is -- yes, with the same coauthor as SICP. It's about math more than CS, and it's really good: by the last chapter you're figuring out motion in general relativity. It has a spirit of DIY exploration unlike any other math book I'd seen at the time.
Something more than a weekend splurge:

Turtle Geometry: The Computer as a Medium for Exploring Mathematics

...and I've not yet tried it, but netlogo sounds interesting.

Offtopic: That netlogo site presents what appears to be an Internet2 cert signed by "InCommon RSA Server CA". It's untrusted by Firefox 45.0.2.
Turtle Geometry book as a "borrowable" ePub (DRM).
Apr 07, 2014 · vinalia on Teaching Devina to Code
It might be fun to look at LOGO (maybe UCBLogo[1], free books included) for a first programming language. This has a first-person (turtle) view on a GUI that you move around to make shapes and do math/physics. The idea is that when programming it will be easier for the programmer to associate themselves with the turtle and interaction/exploration in the language will be natural.

The Logo way is pretty different from conventional programming models because it was tailored to be more intuitive than conventional languages like C, JavaScript, or VB. It still offers access to complex, higher order programming concepts like algorithms, AI, automata, etc. Harold Abelson from MIT (SICP) wrote a cool book that covers math/physics in Logo, too.[2]

The creator of the language has an awesome book[3] on how computers can enhance pedagogy and someone wrote a cool blog post on programming for children that mentioned it too[4].





Apr 07, 2013 · sea6ear on Spirograph in HTML 5
I find Hal Abelson's book Turtle Geometry [1] fascinating. It describes a dialect of Logo (the graphical routines could probably be implemented relatively easily in Python/Tkinter or Tcl/Tk or something like Processing).

Once the basics are described, then it uses them to go on to explore things like non-euclidian geometry (and maybe topology?).


Learning to build or repair a car would probably improve your understanding of thermodynamics, aerodynamics, momentum, etc. Likewise, writing a computer program that simulates the motion of a planet around a star or renders 3D graphics might improve your understanding of classical mechanics and any number of topics in math, just to name a few examples; cf.

This is not to mention that learning how to program a computer is just another tool to put in your bags of tricks for solving problems in any of the domains you mentioned (some better suited than others, of course).

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