1.     Introduction - One picture is worth a thousand words, or is it?  

In their struggle towards obtaining a sense of meaning for the hypercube (a 4-d cube), Davis & Hersh describe vividly, the important role played by manipulating a visual representation:

 "...I was impressed by ... the sheer visual pleasure of watching it. But I was disappointed;

I didn't gain any intuitive feeling for the hypercube... I tried turning the hypercube around, moving it away, bringing it up close, turning it around another way. Suddenly I could feel it!

The hypercube had leaped into palpable reality, as I learned how to manipulate it..." (Davis & Hersch, 1981).

* * *

"One picture is worth a thousand words". Underlying this well-known saying is the widespread experience that a relatively simple visual representation can replace a lot of, and save hours of talking.

The impact of visual representations on understanding, and even more so, on actively doing mathematics, has been intensively researched in the past decades and is widely recognized nowadays.

This is particularly true for the study of 3-d geometry (e.g., Parzysz, 1999; Barwise & Etchemendy, 1991; Zimmermann & Cunningham, 1991; ).

Actually, the employment of visual representations in the study of spatial geometry comes very natural and handy.

It is commonly agreed that 2-d drawings and 3-d concrete models are necessary tools, which provide comprehensive understanding of 3-d configurations.

What about the role of verbal descriptions? Are they necessarily needed? Simply redundant? Or are they inappropriate, may be even disturbing???

In this presentation we attempt at demonstrating the crucial role verbalization plays as a complementary mode to visualization, for dealing with 3-d geometry tasks, understandably and insightfully.

Neither visualization alone nor verbalization in itself suffice for meaningful conceptualization.

This claim is supported by Kosslyn (1980) and by Stigler (1984) who suggested that visual and verbal representations of the same information have distinct attributes augmenting one another.

To elaborate on the issue we focus our attention to the elementary processes of watching a solid or a 3-d configuration, followed by articulating its structure and properties.

We'll show that the latter, promotes insight and understanding which might not be gained  through sheer observation nor by manipulating the visual representations.

This we do in the context of convex Deltahedra - - Polyhedra with equilateral triangular faces.

* * *

To conclude this introduction, here is a personal anecdote by the first author:

The three Platonic Polyhedra made out of equilateral triangular faces, were old "acquaintances" when I was first introduced to the set of convex Deltahedra [1].

The deltahedral dipyramids [2], were quite easy to grasp.

However, there were three other Deltahedra, D12, D14 and D16 which remained altogether "strangers".

It was difficult for me to gain any intuitive feel for them - I was confused. I got a hold of them, only as I was able to construct a verbal description for each(!).

"Suddenly I could feel them! They had leaped into palpable reality", just like Davis' experience with manipulating the visual representation of the hypercube.


How many convex Deltahedra possibly exist?


[1]     Deltahedron is a polyhedron with equilateral triangular faces

[2]     Dipyramid is a solid made of two congruent pyramids with a common base