Display Height and Reach: The Reachable Fraction of a Market Stall and a Stepped Display Rule

A flat market table hides much of its own stock. This paper calculates, from stated reach assumptions, what fraction of a flat display lies within a customer's comfortable reach, within a customer's maximum reach, and within a vendor's reach from the opposite side. The customer is assigned a comfortable reach of 40 cm beyond the table edge and a maximum reach of 60 cm. The vendor is assigned 50 cm comfortable and 70 cm maximum. On a flat table 120 cm deep, only 0.3333 of the stock, uniformly distributed by depth, lies within customer comfortable reach, and only 0.5000 lies within customer maximum reach. The middle 30 cm is beyond comfortable reach from either side, although it can be reached at full stretch by someone. On a table 150 cm deep a dead zone appears: 20 cm in the middle is beyond maximum reach from either side. Raising the display surface makes matters worse, with comfortable reach falling by 4 cm for each 10 cm the surface rises above 95 cm. A sloped board does not by itself solve the problem, because reach along the slope is still the limiting distance. The paper therefore originates a stepped display rule: three tiers, each 32 cm deep, each step 18 cm high, with a clearance penalty of 6 cm per lower tier in the reach calculation. That design has a total depth of 96 cm and places 0.9583 of its stock within customer comfortable reach and 1.0000 within customer maximum reach, an improvement of 62.5 percentage points in the comfortable fraction over the 120 cm flat table, while the vendor reaches the whole design comfortably from the back. The rule for general use is to size each tier at or below 32 cm, keep the number of tiers at three under the stated clearances, and treat any flat depth beyond 90 cm as a display that requires either vendor service for most of its stock or a stepped rebuild.