arXiv:2610.36590v1 Announce Type: new Abstract: Scatter plots draw one mark per observation, so observations that share a location are hidden. In a two-item Likert survey of 2,757 respondents, 98.7% of the data lie under other points and the plot shows only 36 dots. The density-distribution sunflower plot of Dupont and Plummer addresses this by drawing raw points in sparse regions and sunflowers, whose petals count observations, in dense ones. It is not available in R, and its single petal weight for dense cells fails when cell counts span more than an order of magnitude: small dense cells collapse to one petal, and the count a reader decodes can be wrong by 150% or more. We propose geometric petal tiers, in which each cell uses the smallest weight among 1, b, b2, . . . observations per petal that keeps its flower within P petals. Every flower then has between about P/b and P petals, and the relative error of every decoded count is below b/(2P), 17.9% at the defaults b = 5 and P = 14. We also define tie diagnostics and decision rules that use them to choose between an ordinary scatter plot, jitter or transparency, a classic sunflower plot and geometric tiers. Both are implemented in the ggplot2 extension ggsunflower. We illustrate the methods on survey, automotive, historical and large commercial data, and in a simulation over 35 conditions geometric tiers keep the worst-cell error at or below 14% where a single weight reaches a median of 251%.