Listen To The Experts.

             Classic toy.  Full-spectrum cognitive tool (development and research).  


“As a child development specialist, I wanted to let you know how enormously impressed I am with the S3.

“I was particularly taken with the possibilities the S3 offers for strengthening spatial, mathematical, and problem-solving skills in children, beginning in the preschool years.  The S3 readily engages and captivates children, sustains their attention, and challenges and stretches them cognitively.  Moreover, it enables children to calibrate their own complexity of play--making the S3 ideal for any age and age-mix of players, child-child or child-adult.  We have much scientific evidence to indicate that children benefit enormously from such collaborative endeavors, both cognitively and socially.”  (May 13, 2008, cc m.s.)

“While all children would benefit from from play with the S3, its potential contribution to the cognitive development of girls interests me most. We now have substantial research evidence indicating that females lag behind males in certain spatial abilities beginning in early childhood and persisting throughout the life span—a gender gap that contributes to wide-ranging differences in cognitive capacities and opportunities, from everyday map-reading to entry into careers in physical and engineering sciences.  We also know that spatial skills respond readily to relevant experiences, with improvements in performance often larger than the sex differences themselves. The S3 is an ideal context for providing such experiences.

“The S3 is also a wonderful means of stimulating adult-child and child-child engagement and in an open-ended, mentally stimulating, endlessly fascinating pursuit.  Children benefit enormously-from such collaborative endeavors, both cognitively and socially."

Laura E. Berk, Ph.D.  --  University Distinguished Professor

Psychology, Illinois State University 

Child Development, 8th Edition


"I'd like to see the S3 in every preschool and elementary school classroom, and in the home of every family with young children.”  (April 10, 2008, cc m.s.)


                                   S3 Day at Pennsylvania Academy of the Fine Arts -- April 9, 2002  


“I am writing to endorse the Sommer Cube (S3), designed by Michael Sommer, as a worthwhile mathematical toy.  It has all the desirable features of an educational toy, in that 

(a) it instantly appeals to experimenting kinds of activities; 

(b) it motivates children (and adults) to ask, and solve, questions about the 3-dimensional world we live in;

 (c) it lends itself to creative play either alone, or in groups (young and old).

“Although I haven't actually had the opportunity to hold these cubes in my hand I would hope they have a satisfying feel to them -- that is they "feel good" when you hold them. This is a very important feature for toys that involve the handling of various parts.  Two bad examples, where the "feel right" feature isn't present, that I've seen recently used in public schools are:  i) playing Mancala (a pits and pebbles game) with egg cartons and beans (instead of using wooden playing boards with polished stones) and ii) trying to play card games with marked index cards (instead of real cards that slide against each other).   

“In both of these cases the students soon lost interest because they couldn't manipulate the pieces easily.   This is a feature that consumers should always take into account when buying manipulative toys.  The objects, themselves, should be satisfying to hold.  

“I hope that the Sommer Cube will be made available on a wide basis, not only to educators, but to the general public.  This toy definitely has the potential to help anyone who plays with it develop better spatial awareness, and to foster problem-posing and problem-solving.”    (July 7, 2008, to m.s.)

Jean Pedersen, Distinguished Professor  --  Mathematics & Computer Science

Santa Clara University, Santa Clara, CA

Mathematical Reflections -- In a Room with Many Mirrors, 2nd Edition 

Nathalie Sinclair,  Mathematics and Beauty: Aesthetic Approaches to Teaching Children 

"beautiful. playful. and timely: my 2.5 year old is getting ready to move beyond all those rectilinear wooden blocks!" (August 11, 2007)

Bob Naimi, "ToyDirectory Monthly Magazine"

“This is a wonderful Puzzle / Game.  Is it being sold in toy shops yet?” (Feb 23, 2007)

Norman Brosterman, Inventing Kindergarten

"Fabulous. Who are you? Is anyone going to make these for sale?" (Sep 6, 2007)

Godfried Toussaint, Computational Geometry

"Fantastic idea and toy!" (Jan 30, 2007)

Reuben HershWhat is Mathematics, Really?

"that is beautiful!"  (Mar 16, 2007) .... “a great teaching tool for young kids, I hope it becomes available for schools that want to use it.” (Sept 18, 2008)

Joseph MalkevitchWhat's New In Mathematics, Feature Column AMS

"... a work of beauty that fits into my budget" (Feb 2, 2007) .... “represent for many teachers a tool for encouraging analytical thinking of a wide variety ...” (Sept 18, 2008)

Barbara Tversky, "Visuospatial Reasoning"

"Fun and beautiful at the same time." (Feb 19, 2007)

“Like you, I am a fan of spatial thinking and how to get it… the very promising tools yu’ve developed."  (November 29, 2016)

Stewart BrandThe Whole Earth Catalog

".. cool work" (April 17, 2007)

Ivars PetersonThe Mathematical Tourist

"very cool!" (Jan 31, 2007)

Joe RosenSymmetry Discovered

"I think I wouldn’t mind playing with those cubes" (Feb 14, 2004)

Juanita Copley, The Young Child and Mathematics

"I really love this ... can you tell me how I can get some?" (May 27, 2008)

John LienhardThe Engines of Our Ingenuity: An Engineer Looks at Technology & Culture

"That is truly neat.  Is this a sale item?" (February 3, 2007)

Gottfried Mayer, Editor, "Complexity Digest"

"Looks great. Where can one buy them, How much?" (January 17, 2007)

Greg FredericksonDissections Plane & Fancy,

"It looks like a lot of fun and great mental exercise." (Jan 23, 2006)

Ron GrahamConcrete Mathematics:  A Foundation for Computer Science, 2nd Edition

"Looks interesting!" (Jul 20, 2007)

Lajos Szilassi,  Szilassi Polyhedron

"It is really interesting for me. :-)" (Jan 31, 2007)

Keith Moffatt, Fellow of the Royal Society, Perspectives in Fluid Dynamics

"It looks nice.  How do I order one?" (Apr 29, 2007)

James Propp, Who Proved Fermat's Theorem?

"... the Sommer Cube does look like the kind of thing I like!  Are you actually manufacturing it, or is it currently just an idea?" (Apr 23, 2007)

Uri Wilensky, "International Journal of Computers for Mathematical Learning"

"These do look interesting.  "I'd like to look at them.  Are they available now?" (Mar 20, 2007)

Neil SloaneSphere Packings, Lattices and Groups, 3rd Edition

"interesting cube!" (March 20, 2007)

Dick Palais, "The Visualization of Mathematics:  Towards a Mathematical Exploratorium"

"looks really neat" (Mar 19, 2007)

Peter, NorvigArtificial Intelligence:  A Modern Approach, 3rd Edition

"Let me know when you have manufactured a set of blocks I can buy." (Feb 26, 2007)

Doris SchattschneiderVisions of Symmetry

"It looks very interesting and also a very frustrating puzzle." (Aug 14, 2007)

Kieran EganThe Educated Mind:  How Cognitive Tools Shape Our Understanding

"Indeed, fascinated!" (May 2, 2008)

Cliff PickoverArchimedes to Hawking:  Laws of Science and the Great Minds Behind Them

"I think my discussion group would love to hear more about your work." (Feb 26, 2007)

"I wonder if I might have permission to tweet this nice photo to my 21,000 Twitter followers, with a link to your Web page, to stir up interest in your page." (Jan 20, 2016)

Jean PedersenMathematical Reflections -- In a Room with Many Mirrors, 2nd Edition

"How do you obtain these fascinating toys?  I would love to get some.”  (Mar 15, 2007)

Laura BerkChild Development, 8th Edition

"It is an amazing toy--very remarkable" (April 10, 2008)

Gordon Burghardt, “Play and the Brain in Comparative Perspective”

“Looks neat.  How can I get one to look at and evaluate?” (January 5, 2009)

Kathy Hirsh-Pasek, Einstein Never Used Flashcards

“Neat.  I would love to learn more and to play with this sometime.” (January 28, 2011)

“You really created something wonderful that deserves to be widely used and seen.” (April 15, 2011)

Marvin MinskyThe Psychology of Computer Vision   

“I can see some interesting problems for arranging such cubes.” (May 26, 2012)

Don NormanThe Design of Everyday Things    

“Sommercubes are indeed fascinating.” (August 26, 2012)  

“yup, your stuff at is fantastic."  (Jan 26, 2016)

Mark Wexler, "Motor Process in Mental Rotation"

Very elegant and interesting idea. What have you done with your cubes?" (March 21, 2013)

Daniel Schwartz, "Tool Use and the Effect of Action on the Imagination"

Very cool.  This has to be useful for all things biological!" (March 26, 2013)

Christian Freska, Spatial Cognition, Volume 2

“We might have use for your system in a future project that we are currently discussing(July 28, 2013)

Simon Baron-Cohen, Autism: The Facts

“We are looking at whether these could be included in our research(October 23, 2013)

Alan Blackwell, "What is programming?" 

"I'm certainly intrigued by devices like this ..."Are you aware of any attempts to create a mapping between your cube system and defined curriculum items?" (November 25, 2013)

Daniel C. DennettThe Intentional Stance, Sixth Printing

"This is a fascinating object.” (April 28, 2014) 

Michael Levine, DirectorJoan Ganz Cooney Center at Sesame Street Workshop

"Your website and the cube are most thoughtful and provocative.”  (May 15, 2014) 

"I appreciate your drive to put children's voices front and center in the Ed reform and tech discussions. We adults certainly have not had a stellar record!” (May 23, 2014)

Alexander Christakis, "The Predicament of Mankind:  Quest for Structured Responses for Growing World-wide Complexities and Uncertainties"

“… very interesting.  I am sharing it with some of my colleagues for further discussion.”   (July 14, 2014)

"Thank you very much for sharing these wonderful ideas about science with me.  I will look at your thinking carefully ….  (April 12, 2017) 

Dirk Baecker, Schlüsselwerke der Systemtheorie

“Quite amazing, indeed.”  (July 21, 2014) 

Lin Kingkong, "The All-in-One Model"

"Maybe the most wonderful thing happened in the ISSS was met you and saw the S3 cube there, and really understood what you want to change the mindset of learning  most powerful tool to frame our new form of system thinking. (8/1/14)

William Gruver, "Distributed Intelligent Systems:  What Makes Them Intelligent"

"I encourage you to submit a paper or organize a session for SMC2015  [Systems, Man, Cybernetics] to be held in Hong Kong next year.” (8 / 20 / 2014) 

 Loet Leydesdorff, A Sociological Theory of Communication

“It is beautiful.”  (November 30, 2014)  

Andy Clark, Supersizing the Mind:  Embodiment, Action, and Cognitive Extension

“Cool stuff … !!”  (December 17, 2014)

Lou Kaufman, Formal Knot Theory

“ … we all understand that “logic is necessary but not sufficient”.  All researchers and inventors know this and work with this.  You have a powerful method of teaching it.  There are others but yours is yours, original and useful and should be framed in the understanding of the need for experience and intuition.” (July 26,, 2015)

"This is an educational system that should be manufactured and made available for people to learn with. (Jan 23, 2016)

 Henry Lieberman”User Interface Goals, AI Opportunities"

"These look like really cool manipulables, they'd be fun for kids and teaching math.”  (December, 18, 2015)

Dharmendra Mocha”Cognitive Computing Building Block:  A Versatile and Efficient Digital Neuron Model for Neurosynaptic Cores"

“Will keep in mind.”  (January, 26, 2016)

Mark Keane,  ”Constraints on Analogical Mapping: A Comparison of Three Models”  

"There is not a lot of work on learning to develop one’s analogical abilities; though exposure to different types of analogical comparison (which seems to be what you are proposing) could help; this however would need to be proven empirically.  In Polya’s primer on improving creativity in maths he proposed doing something similar.  For instance, one of his heuristics was, if you have a problem in x dimensions that you can’t solve, then reduce it to a problem in x-1 dimensions and see if you can solve it there; then go back to X-dimensions with that solution."  (May 19, 2016)    

Ronaldo VigoMathematical Principles of Human Conceptual Behavior

“I think a caveman [without language or symbols] would be able to [manipulate S3 successfully] .… consistent with the subsymbolic character of my GIST (Generalized Invariance Structure Theory; Vigo, 2013, 2014” …. ). Why not run rigorous experiments to find out?”  (June, 8, 2015)

Donald G. Janelle, Space in Mind: Concepts for Spatial Education

"This looks like an exciting initiative … I am copying this to UCSB colleagues … I'm sure they will be intrigued by the possibilities.”  (December 1, 2016) 

“The S3 link [Childrens Museum] would offer a good  grounding in STEM, as you suggest.”  (January 8, 2016) 

Joseph Kerski, Spatial Mathematics: Theory and Practice through Mapping

"Very clever indeed.   Real world applications too …. I wish the pilot carrying Buddy Holly would have been able to read his instrumentation better.”  (December 7, 2016) 

Krista Francis, "Robotics and Spatial Reasoning"

“This is awesome! … Have you done any research on these S^3 cubes?”  (December 9, 2016)

John EliotModels of Psychological Space

“I would like to learn more about your papers / activities regarding spatial intelligence …. to acquaint former students and others with your work.”  (January 3, 2017)

Richard Boehm, Building Geography Skills for Life

“I like it from a research and a teaching point of view …. it has enormous promise for some NSF funded research in STEM education”  (January 29, 2017)

Dan Ariely, "Investigation Momentum:  The Rentless Pursuit to Resolve Uncertainty"

This is great.  (February 16, 2017) 

David Krakauer“Simple Connectionist Models of Spatial Learning in Bees"

Yes I agree -- it is important to understand these ideas [Exactly why did Einstein and Frank Lloyd Wright love blocks as children?] …. a galaxy of wonder.  (February 20, 2017) 

Peter GrayPsychology, 7th Edition

It does look like a great toy.  (February 22, 2017)

Lila Gleitman, "Turning the Tables: Language and Spatial Reasoning"

I wish you good fortune in pursuing your interesting ideas and plans.  (April 14, 2017) 

George McCloskey, Essentials of Executive Functions Assessment, 2nd Edition

This is all very intriguing to me.  (April 18, 2017) 

Cheryl E. Prager, “Finite Transitive Permutation Groups and Bipartite Vertex-transitive Graphs"

“It looks a lovely system …. enjoyed looking at the web site.”  (May 21, 2017)    

Steven LeharThe World in Your Head:  A Gestalt View of the Mechanism of Conscious Experience

“Brilliant!  You invented it?  Very elegant!”  (May 26, 2017)

Raymond Pavloski, "Searching for a Neural Isomorph of the Perceptual Gestalt"

“I … am very intrigued.”  (June 2, 2017)    "I have always found Gestalt ideas to be the most interesting and, I think, the most fertile in psychology and related disciplines ... Gestalt work has suffered from a poverty of conceptual tools and from the technical inability to do the kind of research required to reveal the power of its ideas.”   (June 8, 2017)

Ian VerstegenCognitive Iconology:  When and How Psychology Explains Images

“… it seems you are onto something important.   I am very impressed by the topology of Rene Thom and the idea that elementary catastrophes can serve as mathematical formalisms for disparate phenomena.”  (June 12, 2017)

Richard E. Clark, “An Analysis of the Failure of Electronic Media and Discovery-Based Learning"

“I support your urging more focus on STEM subjects and earlier than now planned.  As a cognitive psychologist my concern is how the subjects are being taught - most often with enthusiasm that lacks substantive evidence for support and/or is helpful to only a small number of kids (and adults).”  (June 25, 2017)

Gyorgy Turan, "It Is a Long Way from Input to Output:  A Survey of Computational Learning Theory.

“It’s quite interesting.”  (June 29, 2017)

Mark Runco, Creativity:  Theories and Themes — Research, Development, and Practice, 2nd Edition 

“Quite interesting.”  (July 1, 2017)

Wai-Kang Yeap, "A Computational Theory of Human Perceptual Mapping." 

"HI Michael – thanks; fascinating! Minsky’s connectionist AND symbolic? The write-up is intriguing. Will show it to some of my colleagues.”  (July 4, 2017)

Nicola S. Clayton, “Ways of Thinking:  From Crows to Children and Back Again". 

How fascinating.”  (October 25, 2017)  

Marc Herbert, Leonardo Board Chair.

" it's fascinating and fits well within the art/science areas that interest Leonardo.  It also is highly relevant to the growing STEM to STEAM movement    (January 1, 2018) 

Sheryl A. Sorby, Developing Spatial Thinking.

Very interesting.”  (February 6, 2018)  

Ned Block, Functionalism, Consciousness and Representation 

How can you buy this?.”   (February 8, 2018)  

Richard M. Felder, Teaching and Learning STEM:  A Practical Guide

“Fascinating stuff … I find particularly exciting the idea of facilitating the development of childrens’ generally unrecognized and almost always untapped potential for abstract reasoning."   (February 11, 2018)  

S. M. Kosslyn, Image and Mind

“This does look interesting."   (February 11, 2018) 

David Kirsh, “Embodied Cognition and the Magical Future of Interraction Design"

It’s an interesting task that could definately reward when analyzed and studied rigorously."    (February 11, 2018)   

Wayne Weiten, Psychology:  Themes and Variations, 10th Edition

“I will wrap my mind around it next time I revise my developmental chapter."   (February 15, 2018)

 Dale Power, Association for Behavior Analysis International; Exhibits and Promotions Coordinator 

“Did you find a manufacturer to produce the S3 ?  …   Perhaps there may be an opportunity to assist you with selling the product in our bookstore and / or if you have a book on the subject.   (March 13, 2018)

 Maria Droujkova, Natural Math; Founder and Director  

“Thank you for making a difference for people learning mathematics!.   (March 29, 2018)

Barbara McLanahan, Children’s Museum of the Arts; Executive Director  

“This is fascinating.   (April 14, 2018)

Daniel J. Menelly, Doseum (San Antonio’s Muesum for Kids) ; CEO  

I’m deeply attracted to the sculptural form and the ingenuity of your work … I would be particularly keen to see any videographic footage of learners interacting with your media, particularly in group design or problem solving challenges ... if a learning theorist is studying the cube’s efficacy as a tool for engagement or in the assessment of special reasoning, I’d like to know of the research question, and ideally partner in testing it in our exhibit spaces."




                                       Thanks, SJedi.





Margaret, are you grieving

Over Goldengrove unleaving?

Leaves, like the things of man, you

With your fresh thoughts care for, can you?

Ah! as the heart grows older

It will come to such sights colder

By and by, nor spare a sigh

Though worlds of wanwood leafmeal lie:

And yet you will weep and know why.

Now no matter, child, the name:

Sorrow’s springs are the same.

Nor mouth had, no nor mind, expressed

What heart heard of, ghost guessed;

It is the blight man was born for,

It is Margaret you mourn for.

                                                    Gerard Manley Hopkins, "To A Young Child"


                                                            William M. Hollenback, Jr. / Princeton

                                                              Russell L. Ackoff / Wharton / PENN

                                                           Michael S. Sommer  / Wharton / PENN 








© Michael S. Sommer, Ph.D, 2018