Fragments of Infinity: A Kaleidescope of Math and Art / Edition 1

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A visual journey to the intersection of math and imagination, guided by an award-winning author
Mathematics is right brain work, art left brain, right? Not so. This intriguing book shows how intertwined the disciplines are. Portraying the work of many contemporary artists in media from metals to glass to snow, Fragments of Infinity draws us into the mysteries of one-sided surfaces, four-dimensional spaces, self-similar structures, and other bizarre or seemingly impossible features of modern mathematics as they are given visible expression. Featuring more than 250 beautiful illustrations and photographs of artworks ranging from sculptures both massive and minute to elaborate geometric tapestries and mosaics of startling complexity, this is an enthralling exploration of abstract shapes, space, and time made tangible.
Ivars Peterson (Washington, DC) is the mathematics writer and online editor of Science News and the author of The Jungles of Randomness (Wiley: 0-471-16449-6), as well as four previous trade books.

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Editorial Reviews

Publishers Weekly
What do math equations look like when the numbers are translated into form and the form is rendered in, say, silk, or glass? In Fragments of Infinity: A Kaleidoscope of Math and Art, Ivars Peterson (The Jungles of Randomness), a writer and editor at Science News, studies sculpture inspired by abstract math. Many among this breed appear in plazas and subway stations; others get little visibility, being too minute or fragile. Quasicrystals and hypercubes rendered in glass and metal; lattices transformed into different geometric patterns; M?bius strips made of everything from ribbon to bronze; computer sculpture generators via numerous methods and media, the work examined explores "the beauty of embedded possibility." Helaman Ferguson, Harriet Brisson and William Webber are among the artists represented. 250-plus photos and illus. (Oct.) Copyright 2001 Cahners Business Information.
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Product Details

  • ISBN-13: 9780471165583
  • Publisher: Turner Publishing Company
  • Publication date: 9/19/2001
  • Edition number: 1
  • Pages: 240
  • Product dimensions: 9.25 (w) x 7.50 (h) x 0.53 (d)

Read an Excerpt

This book is about creativity and imagination at the intersection of mathematics and art. It portrays the work of several contemporary mathematicians who are also artists or whose mathematical thoughts have inspired others to create. It provides glimpses of artists enthralled by the unlimited possibilities offered by mathematically guided explorations of space and time. It delves into the endlessly fascinating mysteries of one-sided surfaces, four-dimensional spaces, self-similar structures, and other seemingly bizarre features of modern mathematics.

In 1992 I was invited to present the opening address at a remarkable meeting devoted to mathematics and art, organized by mathematician and sculptor Nat Friedman of the State University of New York at Albany. My invitation to the pathbreaking meeting came about because of articles I had written for Science News highlighting the increasing use of visualization in mathematics, particularly the burgeoning role of computer graphics in illuminating and exploring mathematical ideas, from soap-film surfaces, fractals, and knots to chaos, hyperbolic space, and topological transformations. One of my articles had focused on Helaman Ferguson, a sculptor and mathematician who not only works with computers but also carves marble and molds bronze into graceful, sensuous, mathematically inspired artworks.

Friedman's lively gathering introduced me to many more people who are fascinated by interactions between art and mathematics, and with them, I have attended and participated in subsequent meetings. Many of the artists and mathematicians mentioned in this book belong to this peripatetic tribe of math and art enthusiasts. The tribe's diversity of thought and custom, however, also brings to mind difficult questions of what constitutes mathematical art, what beauty means in that context, and what explicit role, if any, mathematics ought to play in the visual arts. The following chapters offer a glimpse of the ways in which we can stretch our minds to imagine and explore exotic geometric realms. They highlight the processes of creativity, invention, and discovery intrinsic to mathematical research and to artistic endeavor.

The book's title echoes thoughts of the Dutch graphic artist M. C. Escher, who sought to capture the notion of infinity in visual images. In 1959, in his essay "Approaches to Infinity," Escher described the reasoning behind one of his intricately repeating designs, which featured a parade of reptiles, as follows: "Not yet true infinity but nevertheless a fragment of it; a piece of 'the universe of reptiles. '

"If only the plane on which [the tiles] fit into one another were infinitely large, then it would be possible to represent an infinite number of them," he continued. "However, we aren't playing an intellectual game here; we are aware that we live in a material, three-dimensional reality, and we cannot manufacture a plane that extends infinitely in all directions."

Escher's solution to his immediate artistic dilemma was to "bend the piece of paper on which this world of reptiles is represented fragmentarily and make a paper cylinder in such a way that the animal figures on its surface continue to fit together without interruptions while the tube revolves around its lengthwise axis." It was just one of many highly original schemes that Escher devised in his attempts to capture infinity visually. Other artists share this passion (or perhaps obsession) for visualizing creations of the mind, whether theorem or dream, and rendering them in concrete form, and they, too, must overcome limitations of their tools and their place in the natural world to present glimpses or fragments of these conceivable yet elusive realms.

Special thanks go to Helaman Ferguson and Nat Friedman, who introduced me to many of the people who and the ideas that inspired and encouraged my travels in the surprisingly wide and diverse world of mathematical art.

I also wish to thank the following persons for their help in explaining ideas, providing illustrations, or supplying other material for this book: Don Albers, Tom Banchoff, Bob Brill, Harriet Brisson, John Bruning, Donald Caspar, Davide Cervone, Benigna Chilla, Barry Cipra, Brent Collins, John Conway, H. S. M. Coxeter, Erik Demaine, Ben Dickins, Stewart Dickson, Doug Dunham, Claire Ferguson, Mike Field, Eric Fontano, George Francis, Martin Gardner, Bathsheba Grossman, George Hart, Linda Henderson, Paul Hildebrandt, Tom Hull, Robert Krawczyk, Robert Lang, Howard Levine, Cliff Long, Robert Longhurst, Shiela Morgan, Eleni Mylonas, Chris K. Palmer, Doug Peden, Roger Penrose, Charles Perry, Cliff Pickover, Tony Robbin, John Robinson, Carlo Roselli, John Safer, Reza Sarhangi, Doris Schattschneider, Dan Schwalbe, Marjorie Senechal, Carlo Séquin, John Sharp, Rhonda Roland Shearer, Arthur Silverman, John Sims, Clifford Singer, Arlene Stamp, Paul Steinhardt, John Sullivan, Keizo Ushio, Helena Verrill, Stan Wagon, William Webber, Jeff Weeks, and Elizabeth Whiteley. My apologies to anyone I have inadvertently failed to include in the list.

I am grateful to my editors at Science News, Joel Greenberg, Pat Young, and Julie Ann Miller, for allowing me to venture occasionally into topics that didn't always fit comfortably within the purview of newsworthy scientific and mathematical research advances. Some of the material in this book has appeared in a somewhat different form in Science News.

I wish to thank my wife, Nancy, for many helpful suggestions while reviewing the original manuscript. I greatly appreciate the efforts of everyone at Wiley who worked so hard to transform an unwieldy stack of manuscript pages and numerous illustrations in a wide variety of formats into the finished book.

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Table of Contents


1. Gallery Visits.

2. Theorems in Stone.

3. A Place in Space.

4. Plane Folds.

5. Grid Fields.

6. Crystal Visions.

7. Strange Sides.

8. Minimal Snow.

9. Points of View.

10. Fragments.

Further Readings.



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First Chapter

Managing the Environment

Environments send messages. The environment in which the mediation occurs can strongly influence the success of the mediation. This environment includes the physical surroundings that affect people's bodily comfort levels. It also involves some less tangible elements: the parties' relative levels of power, their feelings of safety, and arrangements that convey respect. I recommend that mediators pay close attention to all these details. You may find it helpful to create a reusable checklist of environmental factors so you can systematically evaluate each factor described here and any other environmental factors you encounter in your work in light of what you want to achieve in each mediation.

Examining the Power Balance

There is an admonition that warns, "If you think you will fail, you will be right." Viewing this expression through the lenses of mediation, you can appreciate that actual power and perceived power appear the same to parties. Both are treated as real. As a result, I suggest you discern, appreciate, and deal with power in order to achieve a sense of balance between the parties in the mediation environment.

Look at the factors contributing to the impression of power. Social status is one. You can hear it trickle subtly through the conversations of parties who drop names or who talk about connections to well-known personalities. You can see it telegraphed by expensive clothes. At times you will observe parties wearing costly clothes specifically to intimidate the other side. You may notice garments that delineate social standing or caste in other ways. When you see theseindicators, you can help level the scales by creating a power balance through conversation.

You may not observe an actual blue collar at mediation; however, you may sometimes experience an almost tangible gap in employment status. When nonmanagement and executive parties come to the table, the relative difference in positions or professions often creates a sense of power imbalance. This chasm is particularly obvious during some employment mediations. If you think there may be a need for power balancing because of employment status and you believe you can begin to balance it, in a small way, with dress, check with the parties before mediation. Ascertain the kind of clothing they plan to wear. If a great disparity could affect mediation, nudge parties gently by discussing the value of similar clothing as a message that suggests rapport and common ground. Remember, sometimes small factors make big differences in mediation. Additionally, you will want to make sure that both parties are represented by competent counsel so there is a perceived power balance in this area too.

Parties' educational levels are another contributing factor to perceptions of power. The potential imbalance is clear in situations in which one party is accompanied by counsel and the other is not. The unrepresented party may see the party with educated counsel as having an advantage. As a mediator you must be prudent at these times about both balancing parties' knowledge and remaining neutral. You want to avoid communicating the impression that you are advocating for the unrepresented party.

The perception of power when no counsel is present is more subtle. The parties' educational levels may send different messages about power. Some parties see those with higher education or degrees as having an upper hand. Their confidence may dwindle. Previously acquired knowledge of the mediation process can affect parties' self-assurance too. Accordingly, whether parties are represented by counsel or not, I suggest you review the mediation process with all of them yourself. That way, you ensure they all feel knowledgeable and empowered.

Language facility affects perception and levels of power. Parties who express themselves easily feel comfortable discussing their thoughts and playing at verbal repartee with others. People who lack the confidence or ability to express themselves verbally may feel outmatched in the mediation process. For example, lawyers often forget that they are in a mediation environment, not an adversarial pretrial environment. When they have large, incisive vocabularies, they may go to work to eviscerate parties who use more basic vocabularies. When you are the mediator in such a situation, I recommend that you restate the lawyer's ideas and at the same time deescalate the language. You may also want to caucus briefly with the lawyer to advise her of the less-than-productive effect this type of language wreaks on mediation.

Parties whose native language is not English are subject to feelings of vulnerability in mediations conducted in English. This sense arises out of their frustration at their inability to understand others and express themselves as well in a second language as they do in their native tongue. Assist these parties before mediation by covering the terms they will hear used in the mediation. See that they know what steps the process follows. If they have counsel, ask him to preview with them all material they might hear during mediation. Have a translator available or on site if you know or suspect that one might be needed.

The final factor in determining levels of perceived and real power is the money that is available for legal proceedings. You may find parties with unlimited funds feel more powerful than those with financial limits. The former can more easily afford to go to trial. You may see moneyed parties attempting to coerce other parties as a result of this. If so, I suggest you unearth for the coercing parties the benefits of mediation that go beyond money. Caucus with affluent parties, and ask, "What's important to you in mediation?" and, "What will having that [important thing] do for you?" Collect each person's respective touchstones. Then drop these touchstones into your conversation with the affluent parties. You may be able to propel them to seek resolution in mediation.

Providing for Safety

Safety, like power, is an issue in which both perception and reality play strong roles. For example, the number of people at the table may affect feelings of safety. If there are many more people sitting on one side than on the other, parties with fewer representatives may feel overwhelmed or pressured. If you anticipate an imbalance of this kind, canvass both sides before the mediation session. Suggest to them that the presence of an equal number of people sends a message that supports the spirit of mediation and is conducive to achieving positive results. Suggest that if one party feels pushed into settlement, the final agreement may not be durable or may create powerful feelings of buyer's remorse.

The presence or absence of particular people during mediation can affect both how safe an environment is and how safe it seems. That is the reason some jurisdictions do not send cases involving domestic violence to mediation. These cases are excluded to preclude any opportunity for further violence and to reduce an abused or threatened person's fear of harm. I have attended multi-party mediations in which people have been asked to wait in caucus rooms because some parties would not enter the mediation room unless the people who had allegedly threatened them were physically absent from that room. All parties agreed to this physical separation. Although these arrangements were not ideal, they did produce the effect of a safe environment for all the people who gathered in the mediation room. These arrangements also allowed the mediating parties to work toward some type of solution rather than none at all.

Make sure also that obtrusive people are absent. This too communicates a message of safety. Arrange for this before mediation. Ask mediating parties about people who should be barred. The following models suggest how to ensure the absence of obtrusive or threatening people in several scenarios.

Model 1. During the mediation planning stage, ask both Party A and Party B if anyone should not be present at the mediation. If, for example, Party A responds verbally or nonverbally, flesh out the details. If the perception has no basis in fact but the feelings are real, deal with the perception and feelings appropriately before mediation. They could relate to general fear of the mediation. If the perception seems fact based, discuss the request with Party B. In either case, before finalizing the details for the mediation, get agreement from both Party A and Party B on who is to be present.

Model 2. When parties who are potentially troublesome for Party A appear unexpectedly to support Party B at mediation, keep these new arrivals in a physically separate location and away from the mediation. (It is safest that Party A and the problematic people literally not see each other.) Then caucus with Party B. Explain how it will be counterproductive to the mediation process if the bothersome people stay. You might say, "Since Party A is alone, I believe the process would be more balanced and work best for both parties if you were alone too."

Model 3. When multiple parties appear for both sides at the time of the mediation, ask that Party A and Party B each pick from their respective supporters one person to remain at the mediation. Then ask each party to thank the unselected people for coming and to inform them they are not needed at this time.

In addition to dealing with problematic people who may accompany the parties, you can encourage a feeling of safety by taking into account any party's need for translators and validators. Either or both might be required for parties to feel safe in expressing themselves and executing final agreements.

Parties may also draw on their own resources by bringing talismans for feelings of safety. The symbolism of a talisman is usually meaningful only to them. Still, I suggest you check with all parties before mediation to learn whether they plan to bring materials or objects that could be perceived as a threat to the safety of others. Moreover, you must be sure that no weapons will be present. This applies to ceremonial items that others might perceive as weapons as well. This is not a farfetched concern. Not long ago some people were banned from entering a public place because they wore ceremonial daggers. Although wearing these items was part of a religious rite, the public building had a zero-tolerance ban on any kind of weapon. To the people practicing their religion, the knife was part of a meaningful observance. To others, it suggested a threat to their safety.

Finally, look at the message of safety conveyed by the geographical location of the mediation. Canvass the parties about this location. Ask them about their feelings of safety. If the location is one that might be seen as generally unsafe or dangerous after dark, make other arrangements. Doing so will improve the effectiveness of the mediation. Parties can sleep the night before mediation and arrive rested because they are not worried about a dangerous location. They can concentrate during the mediation because they are not concerned that the mediation might run after sundown.

Establishing Comfort

When you consider the different factors affecting communication during mediation, the parties' comfort should be high on the list. Along with power and safety, comfort moderates the river of communication. Low comfort levels impede the stream. High comfort levels sweep mediation forward. Comfort has both physical and emotional components. By understanding the nonverbal messages communicated by the comfort of the environment, you can strategically moderate communication.

Deciding on the Meeting Location

Be mindful of the messages the mediation environment broadcasts and the frame of mind it invites. If you can influence the selection of the mediation environment, you can affect the messages. Parties might be somewhat uncomfortable in an unfamiliar environment. I suggest you do whatever you can to communicate welcome and control. Small acts can make large differences. Before mediation, send parties maps and directions so they know exactly where to go. Include the general inquiry telephone number of the facility too. Arrange for a site that allows parties the choice of public or private transportation. Invite parties to visit the location before the mediation to become more comfortable about driving the route and meeting at the facility. Tell parties in advance if parking is limited. Propose to counsel that they supply prepaid parking passes for parties. If you know about high prices for parking or food in the area, prepare parties for that fact too. Even if they do not like some of the details you convey, the knowledge gives them a sense of control. That sense establishes a productive mind-set for mediation.

Comfort levels are developed in visible and invisible ways. Twenty-five years ago I made a trip to see a lawyer. His office was located on the second floor. Even though I had visited him there many times, this trip was an eye-opener. This time, I was alone, on crutches, and could not pull the second-floor door toward me to open it without risking a fall backward down the flight of stairs. I was powerless until another building visitor came up the stairs to help. Today, legislation guarantees certain types of accessibility. Beyond regulations, however, is the message of intention that barriers or lack of barriers can convey. This message can be powerfully positive when accessibility goes beyond legal mandates. When necessary, I suggest you talk to parties about the strategic advantage gained by physical accommodation resulting from willing acts rather than legal compliance. This perception of accommodation is vital if parties are looking toward a feeling of resolution and not just legal settlement. Similarly, I recommend you find a location that goes beyond conformance to minimum legal standards. Locate one that is comfortable for all the people who are coming to the table.

Comfort levels are affected by issues of territoriality. Parties often consider that mediation held at one party's office site hands the hosting party a "home court advantage." Although the environment may be cost free and the ambience hospitable, small signals can create great feelings of discomfort among the visiting parties. Host parties are used to the office layout, decor, odors, and noises. Most significant, host parties are greeted by people they know and who know them. Visiting parties are strangers who literally do not know the way to the bathroom. I strongly suggest you consider not just the monetary savings but the larger costs attached to the visitors' feelings of discomfort when you have the option of mediating on one party's turf. Unless you determine an overwhelming reason to use one party's home ground, engage a space neutral to both parties.

Preparing the Main Room

If you want to know how to prepare the main and the caucus facilities most effectively, ask yourself, "What do I want to achieve?" and, "What must I do to get there?" You will already have canvassed the parties or their representatives for special accommodation needs. You know that parties operate with different preferred systems. You can anticipate other general needs as well. I suggest you consider all these factors so you can create the most effective nonverbal strategies for productive communication.

Begin with the location of the rooms. Ensure there are washrooms, water pitchers (or a water fountain), coffee, tea, soda pop, and juices near mediation and caucus rooms. Arrange to have coffee brewed and waiting. Serving coffee has useful social implications and sets up a strong kinesthetic, visual, and auditory welcome message for many parties. First, you welcome people with a familiar smell (of coffee). Next, you invite nearness and conversation through an environment in which one party pours coffee while standing beside the other. Give parties access to food from a cafeteria or delivery service. Bring candy or fruit and place it on the table. This strategy is appealing visually and kinesthetically, and these sugar-rich foods can also give parties a quick energy boost.

Select room color carefully. Room color affects parties in different ways. The famous Lüscher Color Test (discussed by Ian Scott in his book of the same name) indicates some effects. Reds seem to elicit uneasiness and sometimes aggression. Yellows can bring out feelings of diligence or envy. Blues and green evoke feelings of tranquility and safety. I suggest you also keep in mind Edward Podolsky's observations in The Doctor Prescribes Colors. Podolsky discusses how people sometimes feel cold in blue rooms and warm in yellow rooms even though the temperature may be the same in both rooms.

Parties with a visual preference appreciate artwork on walls. They need neat, clean, and attractive environments. When areas appear messy or disorganized, these people are strongly affected. They feel uncomfortable and lack the ability to concentrate. People with an auditory preference really care about hearing clearly. They have no real preference about how rooms look. They need environments that are free from ringing telephones, noisy conversations, buzzing faxes, and outside traffic noises. Their attention to sound is so acute that most noises disturb them and direct their attention away from mediation. They need a quiet environment to operate most productively. People with a kinesthetic preference do not care about a room's appearance or acoustics. They want chairs that mold to their bodies and table heights that make it easy for them to write. They like the feel of soft carpet under their feet and general coziness. On the practical side, all parties will feel more relaxed and in control after you tell them the location of pay phones, elevators, fire exits, and smoking areas.

I suggest you look for a mediation room that is large enough to accommodate everyone's need for space and that allows parties to move and stretch. Sufficient space is especially relevant for parties with a high visual component because they need an area in front of them for visualizing in order to make sense of communication. People with a kinesthetic preference need to move around. Even people with an auditory preference often have to change their physical position to shift their mental bearings. The ideal mediation room is one that has small nooks and a suggestion of separate areas. In it, parties can cluster to talk privately among themselves without leaving the room. Unfortunately, ideal rooms are not generally available. If you do get the choice of a room with this unusual configuration, take it. Parties can then meet together and separately without breaking the rapport created in the room.

I recommend you mediate in rooms with windows because the windows provide natural light and the rooms feel open and spacious. The windows must have blinds or curtains, however, to prevent interfering glare or distractions. Make sure all lighting fixtures work. Flickering lights distract people with a visual preference. Buzzing lights annoy people with an auditory preference. If you can meet in a room with adjustable lighting, you can alter the perceptions and mood of all parties in the room. I use strong lighting to wake up parties. I use softer tones to invite them into altered states of awareness in which they can consider matters literally in another light.

Finally, do your best to arrange for mediation and caucus rooms that have individual temperature controls. Rooms that are too hot invite doziness. Rooms that are too cold prevent parties from focusing on issues. Additionally, you may want to set a comfortable room temperature in the morning and then lower the temperature after lunch, when parties tend to get drowsy because an increased blood flow is temporarily routed to the digestive tract.

Preparing the Caucus Rooms

Caucus rooms provide a safe environment in which parties can find privacy. In these rooms, parties hold conversations among themselves and with mediators. Although caucus rooms may not be occupied as long as the mediation room, they must still accommodate parties' needs.

Before making room assignments, form an idea of which room would be best for each party. At the same time, be aware that parties must perceive equality in the assignments. For example, if there are two caucus rooms and a mediation room, parties automatically want to know why they are assigned to one room and not the other. They wonder if you have a special reason for making these room designations. If the choice is arbitrary, tell them. If you have reasons, let them know those too. Reasons could address the number of people in attendance or accessibility. You might give parties the choice of which rooms they use. This strategy provides them with a sense of control over this element in the larger, less manageable thing called mediation. Once parties go into their respective caucus rooms, you may invite them to stake their claim to that room for the day. That way, they feel they have a base of their own and are not just adrift in the mediation facility.

All caucus rooms must be equipped with materials that facilitate communication. See that they contain paper and colored markers for parties who make sense through sight, calculators for parties who need to figure, and comfortable chairs for parties with a kinesthetic preference. Have tissues in the rooms too, because emotions often surface unexpectedly here. On-site tissues relieve parties of embarrassment and tension. They communicate a clear nonverbal message that crying is an acceptable and normal event that happens during this stressful time.

If you have only one caucus room and a mediation room, pay close attention to alternating parties between these rooms. That way, no party will come to believe there is an advantage to the out-of- sight location. If such a belief does get started, it could create tension and make the caucus process divisive rather than constructive.

Finally, stay alert for the unwanted development of ad hoc, informal caucus areas. These are spots in which mediators and counsel may cluster to talk about the progress of the case without the parties being present. This is done surprisingly often and is not very private. Although this discussion is usually off the record, mediating parties are often startled when they stumble onto these conversations. People are talking about them without their knowledge or presence. The surprise is destructive to the process. More than this, it undermines the neutrality of the mediator and counsel in the eyes of the mediating parties. It can undo any good faith that parties invested in the mediator as a neutral. If you find yourself tempted to participate in these caucuses, stop. Avoid creating or being part of these ad hoc activities.

Arranging Tables and Chairs

I suggest that you arrive at the mediation room well before the parties do. You may want to rearrange the furniture. Table and chair positions, sizes, and shapes can influence your nonverbal strategies. You want to exercise awareness and caution when configur-ing this furniture.

Until rapport is developed with the parties, a face-to-face position might instigate a confrontational message. You will find that the ideal mediation table is round. It seats all parties in a circle. There is no head, or end, of the table. Accordingly, it sends a strong nonverbal message of mediator neutrality and party equality. Conversely, when the mediator sits at the end of a rectangular table, it may send an implied message that the mediator is judging rather than facilitating. So, if the on-site table is long and rectangular, you may decide to sit at one end with participant chairs close to you and positioned at a 45-degree angle facing you. I suggest you insist all parties sit along the sides of the table and not at the other end of the table. If some parties were to occupy that end position, it would convey a visual, nonverbal message of power. That would create a perception of mediation imbalance.

If you do not wish to sit at the head of a long, rectangular table, you can create a more informal setting by sitting on one side in the middle and asking the parties to sit across from you. This positioning invites the parties to talk to you first, avoids confrontation, and actually places the parties in a traditional, collaborative, side-by- side position. When the table is small and rectangular and only two single parties are mediating, parties can sit on either side of the same corner or sit on opposite corners of the table and face you. You begin conversation with them facing you. Either seating plan eliminates a confrontational position. Eventually the angle of the parties in relation to each other can change easily and result in a collaborative, side-by-side position as they listen to you. Then it can change back to the original configuration or to a face-to-face conversational position once they are in discussion with each other.

Chairs are the second major furniture consideration. First, look at their positions. If you are co-mediating, I suggest you and your co-mediator sit side by side. That sends a message of equality and collaboration to the mediating parties. This arrangement also makes it easy for co-mediators to exchange messages unobtrusively. When you are co-mediating, you may be asked to allow observers into the room. (This often happens in mediator training and community-based mediation.) Permit only one observer so mediating parties will not feel outnumbered. Seat the observer slightly behind the active mediators so mediating parties pay attention to each other or to the mediators and not to the observer.

Next look at the mobility, appearance, and height of the chairs. Chairs that move and swivel are ideal for mediation. Once mediation is underway, people can move their chairs unobtrusively, sometimes other-than-consciously. When parties are ready to talk to each other, their bodies automatically shift. If parties are sitting in moveable chairs, the angle of the chair turns toward the other party too. That movement is a key cue that parties are shifting mental positions. You can then be ready to deal with that change. If chairs do not swivel, at some point you may have to intervene. When you observe bodies shift, you can suggest that parties might feel more comfortable turning their chairs slightly so they can talk with each other more easily.

I recommend you position the chairs before parties arrive at the mediation. Make sure that all parties have identical chairs. This creates a perception of equality. If any chairs do not match, then get each person a different chair. When some chairs look alike and others look different and this cannot be changed, put some of the identical chairs on both sides of the table to avoid the impression of team against team. This positioning also sends a nonverbal signal of commonality to the parties (even if the common ground is just a chair). Unless all chairs are identical, be especially careful that the mediator's chair looks different from any party's chair. That way you avoid any other-than-conscious message of alignment between the mediator and the party sitting in the same kind of chair the mediator has.

See if you can get chairs that are adjustable. They can be set to equalize parties' different heights. This is often important to perception. Diminutive parties may have their size equalized by using a chair that is as wide as the other party's chair and that raises up enough that the parties can look each other in the eye. (This is the same illusion created constantly on television news broadcasts to equalize news anchors with different physiques.) Adjustable chairs allow parties to find their individual levels of comfort. Stationary chairs in which people are positioned very low in relation to the table can induce feelings of vulnerability. The party's physical relationship to the table may be the same as it was when the party was a small child and a table was too high. This feeling occurs outside conscious awareness and often drives unpredictable, unadult behavior. Chairs that are higher than other chairs sometimes create perceptions of higher rank (a fact made use of in some military environments). These perceptions cross-file into feelings of powerlessness for those not sitting in the higher chairs. Again, the emotion happens other-than-consciously, but the resulting communication is strong and usually counterproductive to the mediation.

Know the sides of the table on which you want to place your parties. Parties who turn to the left will access their auditory side much of the time. That means they will be relying heavily on logical, analytical thought. Parties who turn to the right side a great deal will often find themselves deep in feelings and emotions. You can use this positioning as an effective strategy if you want less emotion from one party and more from another. If you established before mediation that certain sides are nonproductive for specific parties, locate these people carefully in relationship to each other, their counsel, and you. If you are not in a position to change their locations, you can use other subtle physical, eye, paralanguage, and language cues as alternative ways to assist them to be productive.

Using Charts, Props, and Other Items

I suggest you arrange for a writing board that is erasable (a white-board or blackboard). One especially effective whiteboard medium captures an image of what is marked on the board and can print copies of the material. This tool invites parties to participate in the planning process and immediately gives them visual, auditory, and kinesthetic feedback on their progress. Alternatively, you can use an overhead projector, clear transparencies, and colored markers.

It is useful to have a flip chart as well. I suggest you write and draw on flip charts using various colored markers to differentiate information. Have a pencil available to scribble inconspicuous notes in the corners for later reference. If you use permanent ink markers, which can penetrate to the second page and look messy, use every other flip chart page. If you use dry markers, you may be able to use every page.

Flip charts can become the physical record of the beginnings and evolution of the process and of brainstorming sessions. Some parties may regard a flip chart as the tracking mechanism for the mediation. Others may see it as the record on which parties can rely. Parties feel safe in referring to it because they do not have to ask the mediator or other parties for the information. The notations are on the flip chart already. Also, you can sometimes turn sheets back to remind parties how far they have come in the process. Be aware, however, that looking back has the potential to backfire because it can act as an unproductive reminder of a troublesome, earlier part of the mediation.

If you can make arrangements to have two flip charts, do it. You can place one on the left and one on the right side of the table. Each chart then accesses a different representational system and can get different responses from the parties. With two flip charts you may also be able at times to allow the two parties to use the charts for their own planning and for presenting that planning back to the group. Having two flip charts also gives you another alternative. You can anchor, or attach, one chart to unproductive mediation notations (and the negative associations attached to them), put that chart aside, and then move the second flip chart to the center of the room and use it to record positive results.

Give thought to other props and articles you may want in the room. Bring a (silent) clock to increase the comfort level of parties for whom time is relevant. Place it in a visible but unobtrusive position. That way you acknowledge members of monochronic cultures and give members of polychronic cultures a marker in case they do not wear watches. Bring large crayons to use in the switched hand exercise, in which parties write with their nondominant hand in order to get creative. Bring miniature models of cars and trucks when you are mediating vehicular accidents. The miniature size diminishes fear, puts things in a new perspective for parties, and allows people with a kinesthetic preference to communicate effectively about the accident. In your final check, also make sure you avoid props or objects that might be offensive to parties.

See that the environment has devices on site that might be needed to make final resolution possible. Consider the possible need for a computer. Have a fax at your disposal if you think you might need one. When necessary equipment is not available, agreements may fall through before they can be committed to paper. I recall one mediation in which agreement was reached at 2: 00 A. M. Because of timing issues, counsel and parties wanted to write up the final settlement immediately. The law firm on whose premises the mediation had been held had a networked computer system, but the last person who knew how to log onto it had left at 9: 00 P. M. The system was there but locked off. No one at the mediation had a laptop computer, let alone a printer to produce documents. It was hours later when the very fatigued parties executed one very complicated, longhand version of the settlement. They left hoping they had made no errors.

Considering Other Ambience Issues

Explore other issues of ambience that affect mediation. Consider sounds that might be present. Are jackhammers working on the pavement below the window? Is this a quiet area of an office floor or a room next to ringing faxes, telephones, or elevators? Is it only a thin wall away from a loud-voiced speaker? A noisy environment can be disturbing and embarrassing. I remember holding a pre-mediation caucus in a boardroom that shared a wall with another office. The voice of the lawyer next door boomed through the wall. The lawyer also used his speaker phone all the time. My meeting with the client was disrupted constantly by nonstop shouting and private conversations. The client with whom I was meeting was disturbed both by the noise and the fear that his conversations might be as audible to others as the lawyer's were to him.

Before mediation starts, decide on your policy about communication devices and cellular telephones in the mediation room. I recommend that no-cost telephones be available to parties outside the room. If there is a telephone in the room, have incoming calls redirected. Make outgoing calls available to parties who want to pick up messages during breaks and to parties who need to check with external validating sources before they can move ahead. I recommend that cellular telephones and pagers be turned off. Even the vibration signal must be eliminated, or parties receiving calls will still be disturbed. That personal disruption will be communicated nonverbally to other parties in the room.

The final ambience check I suggest is for air quality. See that environment is smoke free and that there is separate accommodation for smokers if necessary. Pay attention to the environment around the proposed meeting place. Steer away from locations situated near overpowering smells like coffee bean roasters, industrial sulfuric smells, and abattoirs. Also make sure that the area is clean and dust free. Be alert for this particularly in newly renovated buildings, in which construction and plaster dust can be a health issue for some parties. When you are considering premises, ask about all these factors.

Planning for Food

Food requirements vary with the type and length of mediation. Most mediation locations make coffee or tea available before the process begins. This is a nonverbal signal of hospitality. As mentioned before, inviting mediating parties to be near each other while getting refreshments often acts as an icebreaker and encourages small talk. Even if the parties do not converse, at least they are literally in the same room with each other. Make available decaffeinated beverages or juices too. Parties may be tense already, so caffeine is counterproductive for them. You may want to arrange for muffins or bagels, to give people who skip breakfast an energy boost at the start of the mediation.

Some mediators like to have refreshments on the table during the mediation. That is strictly a personal choice. It may lend a hospitable air to the environment. If you know that the mediation will last through the lunch hour, I suggest you have food delivered. Canvass parties first about any medical, social, or religious dietary restrictions. Unless I have a specific reason for wanting parties to be physically separate, I usually suggest they stay on site while lunching. If parties leave, they tend to lose their linkage to any positive action that has taken place. I also recommend making a real effort to have parties eat in the same room, even if caucus rooms are available. Eating is a social (as opposed to adversarial) activity. It can be a benign way to have parties relax together without conflicting. Moreover, the delivery of the food or the choice of what to eat once the food is on the table can be a situation that encourages parties to communicate. A decision about food may be their first agreement, one that starts the path to resolution.

Introducing Music

I have discovered that music is an untapped gold mine in mediation. You can make it an inconspicuous element that raises the beat of activity or lowers it significantly. You can choose the times you want to use each musical piece to accomplish a different purpose.

Different kinds of music elicit a variety of responses. You must be thoughtful about what you want to achieve and mindful of the parties attending. If you have gathered any information regarding ethnic preferences, investigate bringing some of that type of music. If parties come from different backgrounds, select neutral music. Know the sounds you want to hear in the room. Know the effect you want to produce. Be as prepared for this component of mediation as for any other element. Be skillful in your choice. On the one hand your selection could be jarring and could interrupt the process. On the other hand it could create a refreshing pause or lead to an upbeat momentum. Preparation means knowing what music you will potentially use. It means bringing a variety of music selected in advance of the mediation.

Finally, test your sound equipment before mediation. Bring extension cords. Make sure there is an outlet or an alternate energy source such as batteries. The music will take care of the rest.

Conveying Respect

Everything you do to make the environment comfortable and safe conveys respect for the parties to the mediation. Perhaps what people will find particularly respectful, however, is attention to their particular needs for space and an awareness of time as they perceive it.

Meeting General and Cultural Space Requirements

In Part One I asked you to consider people's varying needs for space. Now, let's look more closely at the general and cultural space requirements that parties need fulfilled in order to feel comfortable and respected in the mediation environment.

Start with the strategy that the more space you have, the better. You can usually make the environment of a large room seem more intimate; however, you cannot increase the physical size of a claustrophobic room. A larger room allows for flexibility in chair placement and in the way parties sit in relationship to each other. Larger rooms allow for private discussions. Parties and counsel who need to talk privately are not forced to leave the room and break the rapport.

If you know parties will need to hold business discussions and socialize to break the ice, secure a room that accommodates the social distance they need, about four to twelve feet between individuals. Four feet is the minimum. Less than that--from eighteen to forty-eight inches--forces them into a space meant only for personal interactions. Finally, take into consideration that each party will need to maintain a comfortable distance from the mediator and from the other party yet still have available an intimate zone of six to eighteen inches to whisper highly confidential information to colleagues or counsel.

Make provision for special cultural or religious variations from general North American spacing. In addition to the points I mentioned earlier, some cultures require more space around women to ensure they are not touched. Indeed, some cultures require that women sit in a row behind men at mediation. Canvass all these factors before mediation. If you do not, be prepared to accommodate all these different circumstances when parties arrive.

Considering Attitudes Toward Time

To convey respect through your handling of time, I suggest you appreciate and work with the way parties perceive time. The potential fallout from the differing patterns of people from cultures that are monochronic or polychronic is particularly challenging. One party's lateness can ignite the other party's temper, hamstringing the mediation before it begins. I suggest you have a caucus room ready for parties who arrive on time, so they are not in a position to watch the parties arriving late. Give on-time or early parties tasks to do in premediation caucuses. That way, even if polychronic parties are late, monochronic parties do not take offense. They perceive the tasking as the way activities were slated to run.

Be sure parties understand that breaks and meals are scheduled at tentative times only, because once an exact timetable has been put forward monochronic parties will fight to stay on it, even if major progress is being made at that moment. If you stick to fixed times and interrupt a productive session, polychronic parties will find that disrespectful. If you set no times at all, monochronic parties will find that disrespectful.

Finally, assign mediation tasks to parties in ways that respect their views of time. Make sure you give people from a monochronic culture one thing to do at a time and assign multiple tasks to people who come from a polychronic culture.

Checking for Symbolism

When you read briefs and talk to parties prior to mediation, you may expose information that alerts you to possible problems resulting from symbols that are important to one or both of the parties. When you uncover this information, talk to the parties about it before mediation. You can defuse many situations this way. For example, if you know mediation parties come from gangs, you can count on their wanting to wear their gang colors. You must act before mediation. Advise all parties that they must show up at the mediation location without gang identification or they cannot be part of the process.

Get curious about information you receive before mediation. Ask questions. For example, if you are conducting an estate mediation, you might want to inquire about mementos or jewelry that people plan to wear on mediation day. Are there items that could be a red flag to other parties? If you hear about articles that may be problematic, advise parties to leave them at home that day.

Stay alert for other symbolism that may be less conspicuous but just as influential. For example, if mediation is slated to be held in a religious environment, some parties may feel awkward. If it is their own house of worship, they may not like experiencing this type of process in it. If it is not their own house of worship, they may feel uneasy and at a disadvantage. Canvass parties in advance to make sure environments associated with religions or other specific groups are acceptable. If a party objects, find an alternate location.

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