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This paper proposes a method for constructing a 3D model with mechanical properties by embedding multiple haptic maps into an ordinary 2D image. The haptic information described in this paper consists of the depth, stiffness, and viscous damping distribution, each of which takes the form of a 2D grayscale image. After the process of map compression, each instance of haptic information is embedded into a single 2D image via an information hiding method. The proposed 2D image provides multimodal haptic and visual information pertaining to the objects; therefore, a user can experience different haptic sensations according to the extracted haptic information at the point of contact. As the proposed media also offers a method to modify the meshes during interaction, media audiences can participate in the process of tangible media creation. For the selective assignment of mechanical properties, the use of an object-masking map that contains information related to individual object IDs is proposed in the application section. As the proposed image remains compatible with an ordinary 2D image, it can be circulated among existing platforms while preserving its basic functionality as an image. Thus, it is expected that the proposed media can change the manner in which visiohaptic media is consumed by providing realistic haptic information to general users in contrast to other proposals.